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			<H1 id="firstHeading" class="firstHeading" lang="en">Equations of motion</H1>
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<DIV style="padding-top:0.2em;line-height:1.2em;font-size:90%;padding:0.6em 0;font-style:italic;"><A href="/wiki/Second_law_of_motion" class="mw-redirect" title="Second law of motion">Second law of motion</A></DIV>
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<LI><SPAN class="nowrap"><A href="/wiki/Inertia" title="Inertia">Inertia</A> / <A href="/wiki/Moment_of_inertia" title="Moment of inertia">Moment of inertia</A></SPAN></LI>
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<LI><A href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</A></LI>
<LI><A href="/wiki/Appell%27s_equation_of_motion" title="Appell's equation of motion">Appell's equation of motion</A></LI>
<LI><A href="/wiki/Udwadia%E2%80%93Kalaba_equation" title="Udwadia–Kalaba equation">Udwadia–Kalaba equation</A></LI>
<LI><A href="/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics" title="Koopman–von Neumann classical mechanics">Koopman–von Neumann mechanics</A></LI>
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<LI><A class="mw-selflink selflink">Equations of motion</A></LI>
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<LI><A href="/wiki/Mechanics_of_planar_particle_motion" title="Mechanics of planar particle motion">Mechanics of planar particle motion</A></LI>
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<LI><A href="/wiki/Motion_(physics)" title="Motion (physics)">Motion</A> (<A href="/wiki/Linear_motion" title="Linear motion">linear</A>)</LI>
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<LI><A href="/wiki/Simple_harmonic_motion" title="Simple harmonic motion">Simple harmonic motion</A></LI>
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<LI><A href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</A></LI>
<LI><A href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Laplace</A></LI>
<LI><A href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</A></LI>
<LI><A href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</A></LI>
<LI><A href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</A></LI>
<LI><A href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</A></LI>
<LI><A href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Cauchy</A></LI>
</UL>
</DIV>
</DIV>
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<P>In <A href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</A>, <B>equations of motion</B> are <A href="/wiki/Equation" title="Equation">equations</A> that describe the behaviour of a <A href="/wiki/Physical_system" title="Physical system">physical system</A> in terms of its <A href="/wiki/Motion_(physics)" title="Motion (physics)">motion</A> as a <A href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</A> of time.<SUP id="cite_ref-Physics_1991_1-0" class="reference"><A href="#cite_note-Physics_1991-1">[1]</A></SUP> More specifically, the equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables: normally spatial coordinates and time are used, but others are also possible, such as <A href="/wiki/Momentum" title="Momentum">momentum</A> components and time. The most general choice are <A href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</A> which can be any convenient variables characteristic of the physical system.<SUP id="cite_ref-Analytical_Mechanics_2008_2-0" class="reference"><A href="#cite_note-Analytical_Mechanics_2008-2">[2]</A></SUP> The functions are defined in a <A href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</A> in <A href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</A>, but are replaced by <A href="/wiki/Curved_space" title="Curved space">curved spaces</A> in <A href="/wiki/Theory_of_relativity" title="Theory of relativity">relativity</A>. If the <A href="/wiki/Dynamics_(mechanics)" title="Dynamics (mechanics)">dynamics</A> of a system is known, the equations are the solutions to the <A href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</A> describing the motion of the dynamics.</P>
<P>There are two main descriptions of motion: dynamics and <A href="/wiki/Kinematics" title="Kinematics">kinematics</A>. Dynamics is general, since momenta, <A href="/wiki/Force" title="Force">forces</A> and <A href="/wiki/Energy" title="Energy">energy</A> of the <A href="/wiki/Particles" class="mw-redirect" title="Particles">particles</A> are taken into account. In this instance, sometimes the term refers to the differential equations that the system satisfies (e.g., <A href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</A> or <A href="/wiki/Euler%E2%80%93Lagrange_equations" class="mw-redirect" title="Euler–Lagrange equations">Euler–Lagrange equations</A>), and sometimes to the solutions to those equations.</P>
<P>However, kinematics is simpler as it concerns only variables derived from the positions of objects, and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the <A href="#Constant_translational_acceleration_in_a_straight_line">SUVAT equations</A>, arising from the definitions of kinematic quantities: displacement (<SPAN class="texhtml"><I>s</I></SPAN>), initial velocity (<SPAN class="texhtml"><I>u</I></SPAN>), final velocity (<SPAN class="texhtml"><I>v</I></SPAN>), acceleration (<SPAN class="texhtml"><I>a</I></SPAN>), and time (<SPAN class="texhtml"><I>t</I></SPAN>).</P>
<P>Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are <A href="/wiki/Translation_(physics)" class="mw-redirect" title="Translation (physics)">translations</A>, <A href="/wiki/Rotational_motion" class="mw-redirect" title="Rotational motion">rotations</A>, <A href="/wiki/Oscillation" title="Oscillation">oscillations</A>, or any combinations of these.</P>
<P>A differential equation of motion, usually identified as some <A href="/wiki/Physical_law" title="Physical law">physical law</A> and applying <A href="/wiki/Defining_equation_(physics)" title="Defining equation (physics)">definitions</A> of <A href="/wiki/Physical_quantities" class="mw-redirect" title="Physical quantities">physical quantities</A>, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the <A href="/wiki/Initial_value_problem" title="Initial value problem">initial values</A>, which fixes the values of the constants.</P>
<P>To state this formally, in general an equation of motion <SPAN class="texhtml"><I>M</I></SPAN> is a <A href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</A> of the <A href="/wiki/Position_(vector)" title="Position (vector)">position</A> <SPAN class="texhtml"><B>r</B></SPAN> of the object, its <A href="/wiki/Velocity" title="Velocity">velocity</A> (the first time <A href="/wiki/Derivative" title="Derivative">derivative</A> of <SPAN class="texhtml"><B>r</B></SPAN>, <SPAN class="texhtml"><B>v</B> = <SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>d</I><B>r</B></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>dt</I></SPAN></SPAN></SPAN>), and its acceleration (the second <A href="/wiki/Derivative" title="Derivative">derivative</A> of <SPAN class="texhtml"><B>r</B></SPAN>, <SPAN class="texhtml"><B>a</B> = <SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>d</I><SUP>2</SUP><B>r</B></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>dt</I><SUP>2</SUP></SPAN></SPAN></SPAN>), and time <SPAN class="texhtml"><I>t</I></SPAN>. <A href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vectors</A> in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in <SPAN class="texhtml"><B>r</B></SPAN> is a second order <A href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ordinary differential equation</A> (ODE) in <SPAN class="texhtml"><B>r</B></SPAN>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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        <mi>M</mi>
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          <mo>[</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">r</mi>
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          <mo stretchy="false">(</mo>
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          <mo stretchy="false">)</mo>
          <mo>,</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mrow class="MJX-TeXAtom-ORD">
              <mover>
                <mi mathvariant="bold">r</mi>
                <mo mathvariant="bold">˙<!-- ˙ --></mo>
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            </mrow>
          </mrow>
          <mo stretchy="false">(</mo>
          <mi>t</mi>
          <mo stretchy="false">)</mo>
          <mo>,</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mrow class="MJX-TeXAtom-ORD">
              <mover>
                <mi mathvariant="bold">r</mi>
                <mo mathvariant="bold">¨<!-- ¨ --></mo>
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            </mrow>
          </mrow>
          <mo stretchy="false">(</mo>
          <mi>t</mi>
          <mo stretchy="false">)</mo>
          <mo>,</mo>
          <mi>t</mi>
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        <mo>=</mo>
        <mn>0</mn>
        <mspace width="thinmathspace"></mspace>
        <mo>,</mo>
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    <annotation encoding="application/x-tex">{\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/969c32e88d55826a347715bee49544f4ab6bf67a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:25.311ex; height:2.843ex;" alt="M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,," /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>t</I></SPAN> is time, and each overdot denotes one <A href="/wiki/Time_derivative" title="Time derivative">time derivative</A>. The <A href="/wiki/Initial_value_problem" title="Initial value problem">initial conditions</A> are given by the <I>constant</I> values at <SPAN class="texhtml"><I>t</I> = 0</SPAN>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
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        <mo stretchy="false">(</mo>
        <mn>0</mn>
        <mo stretchy="false">)</mo>
        <mspace width="thinmathspace"></mspace>
        <mo>,</mo>
        <mspace width="1em"></mspace>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
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        <mo stretchy="false">(</mo>
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        <mo stretchy="false">)</mo>
        <mspace width="thinmathspace"></mspace>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f67a03428ae17cc7974c0bd2a31619004635f9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.264ex; height:2.843ex;" alt="\mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,." /></SPAN></DD>
</DL>
<P>The solution <SPAN class="texhtml"><B>r</B>(<I>t</I>)</SPAN> to the equation of motion, with specified initial values, describes the system for all times <SPAN class="texhtml"><I>t</I></SPAN> after <SPAN class="texhtml"><I>t</I> = 0</SPAN>. Other dynamical variables like the <A href="/wiki/Momentum" title="Momentum">momentum</A> <SPAN class="texhtml"><B>p</B></SPAN> of the object, or quantities derived from <SPAN class="texhtml"><B>r</B></SPAN> and <SPAN class="texhtml"><B>p</B></SPAN> like <A href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</A>, can be used in place of <SPAN class="texhtml"><B>r</B></SPAN> as the quantity to solve for from some equation of motion, although the position of the object at time <SPAN class="texhtml"><I>t</I></SPAN> is by far the most sought-after quantity.</P>
<P>Sometimes, the equation will be <A href="/wiki/Linear_differential_equation" title="Linear differential equation">linear</A> and is more likely to be exactly solvable. In general, the equation will be <A href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">non-linear</A>, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show <A href="/wiki/Chaos_theory" title="Chaos theory">chaotic</A> behavior depending on how <I>sensitive</I> the system is to the initial conditions.</P>
<P></P>
<DIV id="toc" class="toc">
<DIV class="toctitle">
<H2>Contents</H2>
<SPAN class="toctoggle"> [<A role="button" tabindex="0" class="togglelink">hide</A>] </SPAN></DIV>
<UL>
<LI class="toclevel-1 tocsection-1"><A href="#History"><SPAN class="tocnumber">1</SPAN> <SPAN class="toctext">History</SPAN></A></LI>
<LI class="toclevel-1 tocsection-2"><A href="#Kinematic_equations_for_one_particle"><SPAN class="tocnumber">2</SPAN> <SPAN class="toctext">Kinematic equations for one particle</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-3"><A href="#Kinematic_quantities"><SPAN class="tocnumber">2.1</SPAN> <SPAN class="toctext">Kinematic quantities</SPAN></A></LI>
<LI class="toclevel-2 tocsection-4"><A href="#Uniform_acceleration"><SPAN class="tocnumber">2.2</SPAN> <SPAN class="toctext">Uniform acceleration</SPAN></A>
<UL>
<LI class="toclevel-3 tocsection-5"><A href="#Constant_translational_acceleration_in_a_straight_line"><SPAN class="tocnumber">2.2.1</SPAN> <SPAN class="toctext">Constant translational acceleration in a straight line</SPAN></A></LI>
<LI class="toclevel-3 tocsection-6"><A href="#Constant_linear_acceleration_in_any_direction"><SPAN class="tocnumber">2.2.2</SPAN> <SPAN class="toctext">Constant linear acceleration in any direction</SPAN></A></LI>
<LI class="toclevel-3 tocsection-7"><A href="#Applications"><SPAN class="tocnumber">2.2.3</SPAN> <SPAN class="toctext">Applications</SPAN></A></LI>
<LI class="toclevel-3 tocsection-8"><A href="#Constant_circular_acceleration"><SPAN class="tocnumber">2.2.4</SPAN> <SPAN class="toctext">Constant circular acceleration</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-2 tocsection-9"><A href="#General_planar_motion"><SPAN class="tocnumber">2.3</SPAN> <SPAN class="toctext">General planar motion</SPAN></A></LI>
<LI class="toclevel-2 tocsection-10"><A href="#General_3D_motion"><SPAN class="tocnumber">2.4</SPAN> <SPAN class="toctext">General 3D motion</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-11"><A href="#Dynamic_equations_of_motion"><SPAN class="tocnumber">3</SPAN> <SPAN class="toctext">Dynamic equations of motion</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-12"><A href="#Newtonian_mechanics"><SPAN class="tocnumber">3.1</SPAN> <SPAN class="toctext">Newtonian mechanics</SPAN></A></LI>
<LI class="toclevel-2 tocsection-13"><A href="#Applications_2"><SPAN class="tocnumber">3.2</SPAN> <SPAN class="toctext">Applications</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-14"><A href="#Analytical_mechanics"><SPAN class="tocnumber">4</SPAN> <SPAN class="toctext">Analytical mechanics</SPAN></A></LI>
<LI class="toclevel-1 tocsection-15"><A href="#Electrodynamics"><SPAN class="tocnumber">5</SPAN> <SPAN class="toctext">Electrodynamics</SPAN></A></LI>
<LI class="toclevel-1 tocsection-16"><A href="#General_relativity"><SPAN class="tocnumber">6</SPAN> <SPAN class="toctext">General relativity</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-17"><A href="#Geodesic_equation_of_motion"><SPAN class="tocnumber">6.1</SPAN> <SPAN class="toctext">Geodesic equation of motion</SPAN></A></LI>
<LI class="toclevel-2 tocsection-18"><A href="#Spinning_objects"><SPAN class="tocnumber">6.2</SPAN> <SPAN class="toctext">Spinning objects</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-19"><A href="#Analogues_for_waves_and_fields"><SPAN class="tocnumber">7</SPAN> <SPAN class="toctext">Analogues for waves and fields</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-20"><A href="#Field_equations"><SPAN class="tocnumber">7.1</SPAN> <SPAN class="toctext">Field equations</SPAN></A></LI>
<LI class="toclevel-2 tocsection-21"><A href="#Wave_equations"><SPAN class="tocnumber">7.2</SPAN> <SPAN class="toctext">Wave equations</SPAN></A></LI>
<LI class="toclevel-2 tocsection-22"><A href="#Quantum_theory"><SPAN class="tocnumber">7.3</SPAN> <SPAN class="toctext">Quantum theory</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-23"><A href="#See_also"><SPAN class="tocnumber">8</SPAN> <SPAN class="toctext">See also</SPAN></A></LI>
<LI class="toclevel-1 tocsection-24"><A href="#References"><SPAN class="tocnumber">9</SPAN> <SPAN class="toctext">References</SPAN></A></LI>
</UL>
</DIV>
<P></P>
<H2><SPAN class="mw-headline" id="History">History</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=1" title="Edit section: History">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Historically, equations of motion first appeared in <A href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</A> to describe the motion of <A href="/wiki/Mass" title="Mass">massive objects</A>, a notable application was to <A href="/wiki/Celestial_mechanics" title="Celestial mechanics">celestial mechanics</A> to predict the motion of the planets as if they orbit like clockwork (this was how <A href="/wiki/Neptune_(planet)" class="mw-redirect" title="Neptune (planet)">Neptune</A> was predicted before its discovery), and also investigate the <A href="/wiki/Stability_of_the_solar_system" class="mw-redirect" title="Stability of the solar system">stability of the solar system</A>.</P>
<P>It is important to observe that the huge body of work involving kinematics, dynamics and the mathematical models of the universe developed in baby steps – faltering, getting up and correcting itself – over three millennia and included contributions of both known names and others who have since faded from the annals of history.</P>
<P>In antiquity, notwithstanding the success of <A href="/wiki/Priest" title="Priest">priests</A>, <A href="/wiki/Astrology" title="Astrology">astrologers</A> and <A href="/wiki/Astronomy" title="Astronomy">astronomers</A> in predicting solar and lunar <A href="/wiki/Eclipses" class="mw-redirect" title="Eclipses">eclipses</A>, the solstices and the equinoxes of the <A href="/wiki/Sun" title="Sun">Sun</A> and the period of the <A href="/wiki/Moon" title="Moon">Moon</A>, there was nothing other than a set of algorithms to help them. Despite the great strides made in the development of geometry made by Ancient Greeks and surveys in Rome, we were to wait for another thousand years before the first equations of motion arrive.</P>
<P>The exposure of Europe to the collected works by the Muslims of the Greeks, the Indians and the Islamic scholars, such as <A href="/wiki/Euclid" title="Euclid">Euclid</A>’s <I><A href="/wiki/Euclid%27s_Elements" title="Euclid's Elements">Elements</A></I>, the works of <A href="/wiki/Archimedes" title="Archimedes">Archimedes</A>, and <A href="/wiki/Al-Khw%C4%81rizm%C4%AB" class="mw-redirect" title="Al-Khwārizmī">Al-Khwārizmī</A>'s treatises <SUP id="cite_ref-3" class="reference"><A href="#cite_note-3">[3]</A></SUP> began in Spain, and scholars from all over Europe went to Spain, read, copied and translated the learning into Latin. The exposure of Europe to <A href="/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</A> and their ease in computations encouraged first the scholars to learn them and then the merchants and invigorated the spread of knowledge throughout Europe.</P>
<P>By the 13th century the universities of Oxford and Paris had come up, and the scholars were now studying mathematics and philosophy with lesser worries about mundane chores of life—the fields were not as clearly demarcated as they are in the modern times. Of these, compendia and redactions, such as those of <A href="/wiki/Johannes_Campanus" class="mw-redirect" title="Johannes Campanus">Johannes Campanus</A>, of Euclid and Aristotle, confronted scholars with ideas about infinity and the ratio theory of elements as a means of expressing relations between various quantities involved with moving bodies. These studies led to a new body of knowledge that is now known as physics.</P>
<P>Of these institutes <A href="/wiki/Merton_College" class="mw-redirect" title="Merton College">Merton College</A> sheltered a group of scholars devoted to natural science, mainly physics, astronomy and mathematics, of similar in stature to the intellectuals at the University of Paris. <A href="/wiki/Thomas_Bradwardine#Science" title="Thomas Bradwardine">Thomas Bradwardine</A>, one of those scholars, extended Aristotelian quantities such as distance and velocity, and assigned intensity and extension to them. Bradwardine suggested an exponential law involving force, resistance, distance, velocity and time. <A href="/wiki/Nicole_Oresme" title="Nicole Oresme">Nicholas Oresme</A> further extended Bradwardine's arguments. The <A href="/wiki/Oxford_Calculators" title="Oxford Calculators">Merton school</A> proved that the quantity of motion of a body undergoing a uniformly accelerated motion is equal to the quantity of a uniform motion at the speed achieved halfway through the accelerated motion.</P>
<P>For writers on kinematics before <A href="/wiki/Galileo" class="mw-redirect" title="Galileo">Galileo</A>, since small time intervals could not be measured, the affinity between time and motion was obscure. They used time as a function of distance, and in free fall, greater velocity as a result of greater elevation. Only <A href="/wiki/Domingo_de_Soto" title="Domingo de Soto">Domingo de Soto</A>, a Spanish theologian, in his commentary on <A href="/wiki/Aristotle" title="Aristotle">Aristotle</A>'s <I><A href="/wiki/Physics_(Aristotle)" title="Physics (Aristotle)">Physics</A></I> published in 1545, after defining "uniform difform" motion (which is uniformly accelerated motion) – the word velocity wasn't used – as proportional to time, declared correctly that this kind of motion was identifiable with freely falling bodies and projectiles, without his proving these propositions or suggesting a formula relating time, velocity and distance. De Soto's comments are shockingly correct regarding the definitions of acceleration (acceleration was a rate of change of motion (velocity) in time) and the observation that during the violent motion of ascent acceleration would be negative.</P>
<P>Discourses such as these spread throughout Europe and definitely influenced Galileo and others, and helped in laying the foundation of kinematics.<SUP id="cite_ref-book_4-0" class="reference"><A href="#cite_note-book-4">[4]</A></SUP> Galileo deduced the equation <SPAN class="texhtml"><I>s</I> = <SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;">1</SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</SPAN></SPAN><I>gt</I><SUP>2</SUP></SPAN> in his work geometrically,<SUP id="cite_ref-5" class="reference"><A href="#cite_note-5">[5]</A></SUP> using the <A href="/wiki/Mean_speed_theorem" title="Mean speed theorem">Merton rule</A>, now known as a special case of one of the equations of kinematics. He couldn't use the now-familiar mathematical reasoning. The relationships between speed, distance, time and acceleration was not known at the time.</P>
<P>Galileo was the first to show that the path of a projectile is a <A href="/wiki/Parabola" title="Parabola">parabola</A>. Galileo had an understanding of <A href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal force</A> and gave a correct definition of <A href="/wiki/Momentum" title="Momentum">momentum</A>. This emphasis of momentum as a fundamental quantity in dynamics is of prime importance. He measured momentum by the product of velocity and weight; mass is a later concept, developed by Huygens and Newton. In the swinging of a simple pendulum, Galileo says in <I><A href="/wiki/Two_New_Sciences" title="Two New Sciences">Discourses</A></I><SUP id="cite_ref-6" class="reference"><A href="#cite_note-6">[6]</A></SUP> that "every momentum acquired in the descent along an arc is equal to that which causes the same moving body to ascend through the same arc." His analysis on projectiles indicates that Galileo had grasped the first law and the second law of motion. He did not generalize and make them applicable to bodies not subject to the earth's gravitation. That step was Newton's contribution.</P>
<P>The term "inertia" was used by Kepler who applied it to bodies at rest. (The first law of motion is now often called the law of inertia.)</P>
<P>Galileo did not fully grasp the third law of motion, the law of the equality of action and reaction, though he corrected some errors of Aristotle. With Stevin and others Galileo also wrote on statics. He formulated the principle of the parallelogram of forces, but he did not fully recognize its scope.</P>
<P>Galileo also was interested by the laws of the pendulum, his first observations was when he was a young man. In 1583, while he was praying in the cathedral at Pisa, his attention was arrested by the motion of the great lamp lighted and left swinging, referencing his own pulse for time keeping. To him the period appeared the same, even after the motion had greatly diminished, discovering the isochronism of the pendulum.</P>
<P>More careful experiments carried out by him later, and described in his Discourses, revealed the period of oscillation varies with the square root of length but is independent of the mass the pendulum.</P>
<P>Thus we arrive at <A href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</A>, <A href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</A>, <A href="/wiki/Gottfried_Leibniz" class="mw-redirect" title="Gottfried Leibniz">Gottfried Leibniz</A>, et al.; and the evolved forms of the equations of motion that begin to be recognized as the modern ones.</P>
<P>Later the equations of motion also appeared in <A href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</A>, when describing the motion of charged particles in electric and magnetic fields, the <A href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</A> is the general equation which serves as the definition of what is meant by an <A href="/wiki/Electric_field" title="Electric field">electric field</A> and <A href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</A>. With the advent of <A href="/wiki/Special_relativity" title="Special relativity">special relativity</A> and <A href="/wiki/General_relativity" title="General relativity">general relativity</A>, the theoretical modifications to <A href="/wiki/Spacetime" title="Spacetime">spacetime</A> meant the classical equations of motion were also modified to account for the finite <A href="/wiki/Speed_of_light" title="Speed of light">speed of light</A>, and <A href="/wiki/Curvature_of_spacetime" class="mw-redirect" title="Curvature of spacetime">curvature of spacetime</A>. In all these cases the differential equations were in terms of a function describing the particle's trajectory in terms of space and time coordinates, as influenced by forces or energy transformations.<SUP id="cite_ref-Halliday2004_7-0" class="reference"><A href="#cite_note-Halliday2004-7">[7]</A></SUP></P>
<P>However, the equations of <A href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</A> can also be considered "equations of motion", since they are differential equations of the <A href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</A>, which describes how a quantum state behaves analogously using the space and time coordinates of the particles. There are analogs of equations of motion in other areas of physics, for collections of physical phenomena that can be considered waves, fluids, or fields.</P>
<H2><SPAN class="mw-headline" id="Kinematic_equations_for_one_particle">Kinematic equations for one particle</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=2" title="Edit section: Kinematic equations for one particle">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<H3><SPAN class="mw-headline" id="Kinematic_quantities">Kinematic quantities</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=3" title="Edit section: Kinematic quantities">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:302px;"><A href="/wiki/File:Kinematics.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kinematics.svg/300px-Kinematics.svg.png" width="300" height="181" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kinematics.svg/450px-Kinematics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/98/Kinematics.svg/600px-Kinematics.svg.png 2x" data-file-width="524" data-file-height="317" /></A>
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Kinematic quantities of a classical particle of mass <SPAN class="texhtml"><I>m</I></SPAN>: position <SPAN class="texhtml"><B>r</B></SPAN>, velocity <SPAN class="texhtml"><B>v</B></SPAN>, acceleration <SPAN class="texhtml"><B>a</B></SPAN>.</DIV>
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<P>From the <A href="/wiki/Instantaneous" class="mw-redirect" title="Instantaneous">instantaneous</A> position <SPAN class="texhtml"><B>r</B> = <B>r</B>(<I>t</I>)</SPAN>, instantaneous meaning at an instant value of time <SPAN class="texhtml"><I>t</I></SPAN>, the instantaneous velocity <SPAN class="texhtml"><B>v</B> = <B>v</B>(<I>t</I>)</SPAN> and acceleration <SPAN class="texhtml"><B>a</B> = <B>a</B>(<I>t</I>)</SPAN> have the general, coordinate-independent definitions;<SUP id="cite_ref-Relativity.2C_J.R._Forshaw_2009_8-0" class="reference"><A href="#cite_note-Relativity.2C_J.R._Forshaw_2009-8">[8]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\frac {d\mathbf {r} }{dt}}\,,\quad \mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {r} }{dt^{2}}}\,\!}</annotation>
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</DL>
<P>Notice that velocity always points in the direction of motion, in other words for a curved path it is the <A href="/wiki/Tangent_vector" title="Tangent vector">tangent vector</A>. Loosely speaking, first order derivatives are related to tangents of curves. Still for curved paths, the acceleration is directed towards the <A href="/wiki/Center_of_curvature" title="Center of curvature">center of curvature</A> of the path. Again, loosely speaking, second order derivatives are related to curvature.</P>
<P>The rotational analogues are the "angular vector" (angle the particle rotates about some axis) <SPAN class="texhtml"><B>θ</B> = <B>θ</B>(<I>t</I>)</SPAN>, angular velocity <SPAN class="texhtml"><B>ω</B> = <B>ω</B>(<I>t</I>)</SPAN>, and angular acceleration <SPAN class="texhtml"><B>α</B> = <B>α</B>(<I>t</I>)</SPAN>:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65f29a10a1c8c32e70261c0a01a797dcfa8cfd39" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:32.372ex; height:5.509ex;" alt="{\boldsymbol {\theta }}=\theta {\hat {\mathbf {n} }}\,,\quad {\boldsymbol {\omega }}={\frac {d{\boldsymbol {\theta }}}{dt}}\,,\quad {\boldsymbol {\alpha }}={\frac {d{\boldsymbol {\omega }}}{dt}}\,," /></SPAN></DD>
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<P>where <SPAN class="texhtml"><B>n̂</B></SPAN> is a <A href="/wiki/Unit_vector" title="Unit vector">unit vector</A> in the direction of the axis of rotation, and <SPAN class="texhtml"><I>θ</I></SPAN> is the angle the object turns through about the axis.</P>
<P>The following relation holds for a point-like particle, orbiting about some axis with angular velocity <SPAN class="texhtml"><B>ω</B></SPAN>:<SUP id="cite_ref-9" class="reference"><A href="#cite_note-9">[9]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10c0bafc30f7301ee953138c85c97470dbf0ccd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; margin-right: -0.387ex; width:10.56ex; height:1.676ex;" alt="\mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {r} \,\!" /></SPAN></DD>
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<P>where <SPAN class="texhtml"><B>r</B></SPAN> is the position vector of the particle (radial from the rotation axis) and <SPAN class="texhtml"><B>v</B></SPAN> the tangential velocity of the particle. For a rotating continuum <A href="/wiki/Rigid_body" title="Rigid body">rigid body</A>, these relations hold for each point in the rigid body.</P>
<H3><SPAN class="mw-headline" id="Uniform_acceleration">Uniform acceleration</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=4" title="Edit section: Uniform acceleration">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>The differential equation of motion for a particle of constant or uniform acceleration in a straight line is simple: the acceleration is constant, so the second derivative of the position of the object is constant. The results of this case are summarized below.</P>
<H4><SPAN class="mw-headline" id="Constant_translational_acceleration_in_a_straight_line">Constant translational acceleration in a straight line</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=5" title="Edit section: Constant translational acceleration in a straight line">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<P>These equations apply to a particle moving linearly, in three dimensions in a straight line with constant <A href="/wiki/Acceleration" title="Acceleration">acceleration</A>.<SUP id="cite_ref-Physics_P.M_10-0" class="reference"><A href="#cite_note-Physics_P.M-10">[10]</A></SUP> Since the position, velocity, and acceleration are collinear (parallel, and lie on the same line) – only the magnitudes of these vectors are necessary, and because the motion is along a straight line, the problem effectively reduces from three dimensions to one.</P>
<DL>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v&=at+v_{0}\quad [1]\\\end{aligned}}}</annotation>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r&=r_{0}+v_{0}t+{\tfrac {1}{2}}{a}t^{2}\quad [2]\\\end{aligned}}}</annotation>
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                <mo>−<!-- − --></mo>
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                  <mi>a</mi>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}\quad [5]\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7762dee82fc5e2f6add29cbdd2d77bb6cb8755" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -5.005ex; width:27.15ex; height:11.176ex;" alt="{\displaystyle {\begin{aligned}r&=r_{0}+{\tfrac {1}{2}}\left(v+v_{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\r&=r_{0}+vt-{\tfrac {1}{2}}{a}t^{2}\quad [5]\\\end{aligned}}}" /></SPAN></DD>
</DL>
<P>where:</P>
<UL>
<LI><SPAN class="texhtml"><I>r</I><SUB>0</SUB></SPAN> is the particle's initial <A href="/wiki/Position_(vector)" title="Position (vector)">position</A></LI>
<LI><SPAN class="texhtml"><I>r</I></SPAN> is the particle's final position</LI>
<LI><SPAN class="texhtml"><I>v</I><SUB>0</SUB></SPAN> is the particle's initial <A href="/wiki/Velocity" title="Velocity">velocity</A></LI>
<LI><SPAN class="texhtml"><I>v</I></SPAN> is the particle's final velocity</LI>
<LI><SPAN class="texhtml"><I>a</I></SPAN> is the particle's <A href="/wiki/Acceleration" title="Acceleration">acceleration</A></LI>
<LI><SPAN class="texhtml"><I>t</I></SPAN> is the <A href="/wiki/Time_in_physics" title="Time in physics">time interval</A></LI>
</UL>
<DIV class="NavFrame collapsed" style="position:relative;width:100%;border:none;" id="NavFrame7">
<DIV class="NavHead" style="position:absolute;width:3.2em;background:transparent;"><A class="NavToggle" id="NavToggle7" href="#">[show]</A></DIV>
<DIV style="font-size:100%;height:1.6em;background:transparent;padding-left:2.8em; text-align:right;font-weight:bold;font-size:100%; padding-right:2.5em;">  Derivation</DIV>
<DIV class="NavContent" style="text-align: left; font-size: 100%; padding-left: 3em; display: none;">
<P>Equations [1] and [2] are from integrating the definitions of velocity and acceleration,<SUP id="cite_ref-Physics_P.M_10-1" class="reference"><A href="#cite_note-Physics_P.M-10">[10]</A></SUP> subject to the initial conditions <SPAN class="texhtml"><B>r</B>(<I>t</I><SUB>0</SUB>) = <B>r</B><SUB>0</SUB></SPAN> and <SPAN class="texhtml"><B>v</B>(<I>t</I><SUB>0</SUB>) = <B>v</B><SUB>0</SUB></SPAN>;</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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            <mtr>
              <mtd>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">v</mi>
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                <mi></mi>
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                <mo>=</mo>
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                  <mi mathvariant="bold">a</mi>
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                    <mi mathvariant="bold">v</mi>
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                <mspace width="thinmathspace"></mspace>
                <mo>,</mo>
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                <mo stretchy="false">[</mo>
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            <mtr>
              <mtd>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">r</mi>
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              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mo>∫<!-- ∫ --></mo>
                <mo stretchy="false">(</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">a</mi>
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                <mi>t</mi>
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                <msub>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="bold">v</mi>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
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                </msub>
                <mo stretchy="false">)</mo>
                <mi>d</mi>
                <mi>t</mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mi mathvariant="bold">a</mi>
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                      <msup>
                        <mi>t</mi>
                        <mrow class="MJX-TeXAtom-ORD">
                          <mn>2</mn>
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                      </msup>
                    </mrow>
                    <mn>2</mn>
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                </mrow>
                <mo>+</mo>
                <msub>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="bold">v</mi>
                  </mrow>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
                  </mrow>
                </msub>
                <mi>t</mi>
                <mo>+</mo>
                <msub>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="bold">r</mi>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
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                </msub>
                <mspace width="thinmathspace"></mspace>
                <mo>,</mo>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>2</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
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        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} &=\int \mathbf {a} dt=\mathbf {a} t+\mathbf {v} _{0}\,,\quad [1]\\\mathbf {r} &=\int (\mathbf {a} t+\mathbf {v} _{0})dt={\frac {\mathbf {a} t^{2}}{2}}+\mathbf {v} _{0}t+\mathbf {r} _{0}\,,\quad [2]\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b943e1186a0e4294d3d2f438641386b1ec59f57e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -5.338ex; width:43.504ex; height:11.843ex;" alt="{\begin{aligned}\mathbf {v} &=\int \mathbf {a} dt=\mathbf {a} t+\mathbf {v} _{0}\,,\quad [1]\\\mathbf {r} &=\int (\mathbf {a} t+\mathbf {v} _{0})dt={\frac {\mathbf {a} t^{2}}{2}}+\mathbf {v} _{0}t+\mathbf {r} _{0}\,,\quad [2]\\\end{aligned}}" /></SPAN></DD>
</DL>
<P>in magnitudes,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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            <mtr>
              <mtd>
                <mi>v</mi>
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              <mtd>
                <mi></mi>
                <mo>=</mo>
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                <msub>
                  <mi>v</mi>
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                    <mn>0</mn>
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                <mspace width="thinmathspace"></mspace>
                <mo>,</mo>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>1</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mi>r</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mi>a</mi>
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                      <msup>
                        <mi>t</mi>
                        <mrow class="MJX-TeXAtom-ORD">
                          <mn>2</mn>
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                      </msup>
                    </mrow>
                    <mn>2</mn>
                  </mfrac>
                </mrow>
                <mo>+</mo>
                <msub>
                  <mi>v</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
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                <mi>t</mi>
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                  <mi>r</mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mspace width="thinmathspace"></mspace>
                <mo>.</mo>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>2</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v&=at+v_{0}\,,\quad [1]\\r&={\frac {{a}t^{2}}{2}}+v_{0}t+r_{0}\,.\quad [2]\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/317afd1fe00ba07fdc220d0970a7a95cbb7fe843" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.838ex; width:26.1ex; height:8.843ex;" alt="{\begin{aligned}v&=at+v_{0}\,,\quad [1]\\r&={\frac {{a}t^{2}}{2}}+v_{0}t+r_{0}\,.\quad [2]\\\end{aligned}}" /></SPAN></DD>
</DL>
<P>Equation [3] involves the average velocity <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><B>v</B> + <B>v</B><SUB>0</SUB></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">2</SPAN></SPAN></SPAN>. Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from <SPAN class="texhtml"><B>v</B><SUB>0</SUB></SPAN> to <SPAN class="texhtml"><B>v</B></SPAN>, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows from solving [1] for</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">a</mi>
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        <mo>=</mo>
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          <mfrac>
            <mrow>
              <mo stretchy="false">(</mo>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">v</mi>
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              <mo>−<!-- − --></mo>
              <msub>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">v</mi>
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              <mo stretchy="false">)</mo>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\frac {(\mathbf {v} -\mathbf {v} _{0})}{t}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8623aef168e1d9e1d56b6e30d1e6fe83152c81a8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:13.841ex; height:5.676ex;" alt="\mathbf {a} ={\frac {(\mathbf {v} -\mathbf {v} _{0})}{t}}" /></SPAN></DD>
</DL>
<P>and substituting into [2]</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
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        <mo>=</mo>
        <msub>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">r</mi>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
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        <mo>+</mo>
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          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">v</mi>
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          <mrow class="MJX-TeXAtom-ORD">
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        <mi>t</mi>
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          <mfrac>
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            <mn>2</mn>
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        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">v</mi>
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          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">v</mi>
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          <mrow class="MJX-TeXAtom-ORD">
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        <mo>,</mo>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\frac {t}{2}}(\mathbf {v} -\mathbf {v} _{0})\,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb0a60cbea070552a8e6f457028512692995616e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:27.069ex; height:5.176ex;" alt="\mathbf {r} =\mathbf {r} _{0}+\mathbf {v} _{0}t+{\frac {t}{2}}(\mathbf {v} -\mathbf {v} _{0})\,," /></SPAN></DD>
</DL>
<P>then simplifying to get</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
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        <mo>=</mo>
        <msub>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">r</mi>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
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        </msub>
        <mo>+</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
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            <mn>2</mn>
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        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">v</mi>
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        <msub>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">v</mi>
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          <mrow class="MJX-TeXAtom-ORD">
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        <mo stretchy="false">)</mo>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {r} _{0}+{\frac {t}{2}}(\mathbf {v} +\mathbf {v} _{0})}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f18e723e57b603769e4e5b0854aea595e2edaf86" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:19.841ex; height:5.176ex;" alt="\mathbf {r} =\mathbf {r} _{0}+{\frac {t}{2}}(\mathbf {v} +\mathbf {v} _{0})" /></SPAN></DD>
</DL>
<P>or in magnitudes</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>r</mi>
        <mo>=</mo>
        <msub>
          <mi>r</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
          </mrow>
        </msub>
        <mo>+</mo>
        <mrow>
          <mo>(</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mfrac>
              <mrow>
                <mi>v</mi>
                <mo>+</mo>
                <msub>
                  <mi>v</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
                  </mrow>
                </msub>
              </mrow>
              <mn>2</mn>
            </mfrac>
          </mrow>
          <mo>)</mo>
        </mrow>
        <mi>t</mi>
        <mspace width="1em"></mspace>
        <mo stretchy="false">[</mo>
        <mn>3</mn>
        <mo stretchy="false">]</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle r=r_{0}+\left({\frac {v+v_{0}}{2}}\right)t\quad [3]}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8d0cd441e58cf8f118d4e89eaf8462a45fb02a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:25.654ex; height:6.176ex;" alt="r=r_{0}+\left({\frac {v+v_{0}}{2}}\right)t\quad [3]" /></SPAN></DD>
</DL>
<P>From [3],</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>t</mi>
        <mo>=</mo>
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          <mi>r</mi>
          <mo>−<!-- − --></mo>
          <msub>
            <mi>r</mi>
            <mrow class="MJX-TeXAtom-ORD">
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          <mo>)</mo>
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        <mrow>
          <mo>(</mo>
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            <mfrac>
              <mn>2</mn>
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          <mo>)</mo>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle t=\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acd994aef83b1da3549a9b71a58a8f107a978709" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:22.674ex; height:6.176ex;" alt="t=\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)" /></SPAN></DD>
</DL>
<P>substituting for <SPAN class="texhtml"><I>t</I></SPAN> in [1]:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
            <mtr>
              <mtd>
                <mi>v</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mi>a</mi>
                <mrow>
                  <mo>(</mo>
                  <mi>r</mi>
                  <mo>−<!-- − --></mo>
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                  <mi>v</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
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                </msup>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <msubsup>
                  <mi>v</mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                  <mrow class="MJX-TeXAtom-ORD">
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                <mo>+</mo>
                <mn>2</mn>
                <mi>a</mi>
                <mrow>
                  <mo>(</mo>
                  <mi>r</mi>
                  <mo>−<!-- − --></mo>
                  <msub>
                    <mi>r</mi>
                    <mrow class="MJX-TeXAtom-ORD">
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                  <mo>)</mo>
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                <mn>4</mn>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v&=a\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)+v_{0}\\v\left(v+v_{0}\right)&=2a\left(r-r_{0}\right)+v_{0}\left(v+v_{0}\right)\\v^{2}+vv_{0}&=2a\left(r-r_{0}\right)+v_{0}v+v_{0}^{2}\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a107aebb7f3c3f6d35b60ea640f45683fc17d48" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -7.355ex; margin-bottom: -0.317ex; width:38.797ex; height:16.509ex;" alt="{\begin{aligned}v&=a\left(r-r_{0}\right)\left({\frac {2}{v+v_{0}}}\right)+v_{0}\\v\left(v+v_{0}\right)&=2a\left(r-r_{0}\right)+v_{0}\left(v+v_{0}\right)\\v^{2}+vv_{0}&=2a\left(r-r_{0}\right)+v_{0}v+v_{0}^{2}\\v^{2}&=v_{0}^{2}+2a\left(r-r_{0}\right)\quad [4]\\\end{aligned}}" /></SPAN></DD>
</DL>
<P>From [3],</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mn>2</mn>
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          <mo>−<!-- − --></mo>
          <msub>
            <mi>r</mi>
            <mrow class="MJX-TeXAtom-ORD">
              <mn>0</mn>
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          <mo>)</mo>
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        <mo>−<!-- − --></mo>
        <mi>v</mi>
        <mi>t</mi>
        <mo>=</mo>
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          <mi>v</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>0</mn>
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        <mi>t</mi>
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    <annotation encoding="application/x-tex">{\displaystyle 2\left(r-r_{0}\right)-vt=v_{0}t}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a513b4b2541c336ee184768d6f0c09e4da6e8c76" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.419ex; height:2.843ex;" alt="2\left(r-r_{0}\right)-vt=v_{0}t" /></SPAN></DD>
</DL>
<P>substituting into [2]:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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            <mtr>
              <mtd>
                <mi>r</mi>
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              <mtd>
                <mi></mi>
                <mo>=</mo>
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                <mi>t</mi>
                <mo>+</mo>
                <msub>
                  <mi>r</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
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              </mtd>
            </mtr>
            <mtr>
              <mtd>
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              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mi>a</mi>
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                      <msup>
                        <mi>t</mi>
                        <mrow class="MJX-TeXAtom-ORD">
                          <mn>2</mn>
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                    <mn>2</mn>
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                <mo>+</mo>
                <mi>r</mi>
                <mo>−<!-- − --></mo>
                <msub>
                  <mi>r</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
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                <mo>−<!-- − --></mo>
                <mi>v</mi>
                <mi>t</mi>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mi>r</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <msub>
                  <mi>r</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>0</mn>
                  </mrow>
                </msub>
                <mo>+</mo>
                <mi>v</mi>
                <mi>t</mi>
                <mo>−<!-- − --></mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mfrac>
                    <mrow>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mi>a</mi>
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                      <msup>
                        <mi>t</mi>
                        <mrow class="MJX-TeXAtom-ORD">
                          <mn>2</mn>
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                    <mn>2</mn>
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                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>5</mn>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}r&={\frac {{a}t^{2}}{2}}+2r-2r_{0}-vt+r_{0}\\0&={\frac {{a}t^{2}}{2}}+r-r_{0}-vt\\r&=r_{0}+vt-{\frac {{a}t^{2}}{2}}\quad [5]\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc1c05223bd1aed6159f553e238d24b506cb1d79" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.171ex; margin-top: -0.317ex; width:30.059ex; height:17.509ex;" alt="{\begin{aligned}r&={\frac {{a}t^{2}}{2}}+2r-2r_{0}-vt+r_{0}\\0&={\frac {{a}t^{2}}{2}}+r-r_{0}-vt\\r&=r_{0}+vt-{\frac {{a}t^{2}}{2}}\quad [5]\end{aligned}}" /></SPAN></DD>
</DL>
<P>Usually only the first 4 are needed, the fifth is optional.</P>
</DIV>
</DIV>
<P>Here <SPAN class="texhtml"><I>a</I></SPAN> is <I>constant</I> acceleration, or in the case of bodies moving under the influence of <A href="/wiki/Gravity" title="Gravity">gravity</A>, the <A href="/wiki/Standard_gravity" title="Standard gravity">standard gravity</A> <SPAN class="texhtml"><I>g</I></SPAN> is used. Note that each of the equations contains four of the five variables, so in this situation it is sufficient to know three out of the five variables to calculate the remaining two.</P>
<P>In elementary physics the same formulae are frequently written in different notation as:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
            <mtr>
              <mtd>
                <mi>v</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mi>u</mi>
                <mo>+</mo>
                <mi>a</mi>
                <mi>t</mi>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>1</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mi>s</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mi>u</mi>
                <mi>t</mi>
                <mo>+</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mstyle displaystyle="false" scriptlevel="0">
                    <mfrac>
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                <mi>a</mi>
                <msup>
                  <mi>t</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
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                </msup>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>2</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mi>s</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mstyle displaystyle="false" scriptlevel="0">
                    <mfrac>
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                <mo stretchy="false">(</mo>
                <mi>u</mi>
                <mo>+</mo>
                <mi>v</mi>
                <mo stretchy="false">)</mo>
                <mi>t</mi>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>3</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <msup>
                  <mi>v</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <msup>
                  <mi>u</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
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                <mo>+</mo>
                <mn>2</mn>
                <mi>a</mi>
                <mi>s</mi>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>4</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd>
                <mi>s</mi>
              </mtd>
              <mtd>
                <mi></mi>
                <mo>=</mo>
                <mi>v</mi>
                <mi>t</mi>
                <mo>−<!-- − --></mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mstyle displaystyle="false" scriptlevel="0">
                    <mfrac>
                      <mn>1</mn>
                      <mn>2</mn>
                    </mfrac>
                  </mstyle>
                </mrow>
                <mi>a</mi>
                <msup>
                  <mi>t</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>2</mn>
                  </mrow>
                </msup>
                <mspace width="1em"></mspace>
                <mo stretchy="false">[</mo>
                <mn>5</mn>
                <mo stretchy="false">]</mo>
              </mtd>
            </mtr>
          </mtable>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v&=u+at\quad [1]\\s&=ut+{\tfrac {1}{2}}at^{2}\quad [2]\\s&={\tfrac {1}{2}}(u+v)t\quad [3]\\v^{2}&=u^{2}+2as\quad [4]\\s&=vt-{\tfrac {1}{2}}at^{2}\quad [5]\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8876516f71a06f98f87c759f9df9f4100b1e7072" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.338ex; width:20.729ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}v&=u+at\quad [1]\\s&=ut+{\tfrac {1}{2}}at^{2}\quad [2]\\s&={\tfrac {1}{2}}(u+v)t\quad [3]\\v^{2}&=u^{2}+2as\quad [4]\\s&=vt-{\tfrac {1}{2}}at^{2}\quad [5]\\\end{aligned}}}" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>u</I></SPAN> has replaced <SPAN class="texhtml"><I>v</I><SUB>0</SUB></SPAN>, <SPAN class="texhtml"><I>s</I></SPAN> replaces <SPAN class="texhtml"><I>r</I></SPAN>, and <SPAN class="texhtml"><I>s</I><SUB>0</SUB> = 0</SPAN>. They are often referred to as the <B>SUVAT equations</B>, where "SUVAT" is an <A href="/wiki/Acronym" title="Acronym">acronym</A> from the variables: <SPAN class="texhtml"><I>s</I></SPAN> = displacement (<SPAN class="texhtml"><I>s</I><SUB>0</SUB></SPAN> = initial displacement), <SPAN class="texhtml"><I>u</I></SPAN> = initial velocity, <SPAN class="texhtml"><I>v</I></SPAN> = final velocity, <SPAN class="texhtml"><I>a</I></SPAN> = acceleration, <SPAN class="texhtml"><I>t</I></SPAN> = time.<SUP id="cite_ref-Hanrahan2003_11-0" class="reference"><A href="#cite_note-Hanrahan2003-11">[11]</A></SUP><SUP id="cite_ref-12" class="reference"><A href="#cite_note-12">[12]</A></SUP></P>
<H4><SPAN class="mw-headline" id="Constant_linear_acceleration_in_any_direction">Constant linear acceleration in any direction</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=6" title="Edit section: Constant linear acceleration in any direction">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:402px;"><A href="/wiki/File:Suvat_eom_any_direction_constant_acceleration.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Suvat_eom_any_direction_constant_acceleration.svg/400px-Suvat_eom_any_direction_constant_acceleration.svg.png" width="400" height="204" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Suvat_eom_any_direction_constant_acceleration.svg/600px-Suvat_eom_any_direction_constant_acceleration.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Suvat_eom_any_direction_constant_acceleration.svg/800px-Suvat_eom_any_direction_constant_acceleration.svg.png 2x" data-file-width="597" data-file-height="304" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Suvat_eom_any_direction_constant_acceleration.svg" class="internal" title="Enlarge"></A></DIV>
Trajectory of a particle with initial position vector <SPAN class="texhtml"><B>r</B><SUB>0</SUB></SPAN> and velocity <SPAN class="texhtml"><B>v</B><SUB>0</SUB></SPAN>, subject to constant acceleration <SPAN class="texhtml"><B>a</B></SPAN>, all three quantities in any direction, and the position <SPAN class="texhtml"><B>r</B>(<I>t</I>)</SPAN> and velocity <SPAN class="texhtml"><B>v</B>(<I>t</I>)</SPAN> after time <SPAN class="texhtml"><I>t</I></SPAN>.</DIV>
</DIV>
</DIV>
<P>The initial position, initial velocity, and acceleration vectors need not be collinear, and take an almost identical form. The only difference is that the square magnitudes of the velocities require the <A href="/wiki/Dot_product" title="Dot product">dot product</A>. The derivations are essentially the same as in the collinear case,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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              <mtd>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {a} t+\mathbf {v} _{0}\quad [1]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [2]\\\mathbf {r} &=\mathbf {r} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {v} _{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2\mathbf {a} \cdot \left(\mathbf {r} -\mathbf {r} _{0}\right)\quad [4]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} t-{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [5]\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815d0b024bba8777fc1b0bf021175072d692101d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.204ex; margin-bottom: -0.3ex; width:28.448ex; height:18.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {v} &=\mathbf {a} t+\mathbf {v} _{0}\quad [1]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [2]\\\mathbf {r} &=\mathbf {r} _{0}+{\tfrac {1}{2}}\left(\mathbf {v} +\mathbf {v} _{0}\right)t\quad [3]\\v^{2}&=v_{0}^{2}+2\mathbf {a} \cdot \left(\mathbf {r} -\mathbf {r} _{0}\right)\quad [4]\\\mathbf {r} &=\mathbf {r} _{0}+\mathbf {v} t-{\tfrac {1}{2}}\mathbf {a} t^{2}\quad [5]\\\end{aligned}}}" /></SPAN></DD>
</DL>
<P>although the <A href="/wiki/Torricelli_equation" class="mw-redirect" title="Torricelli equation">Torricelli equation</A> [4] can be derived using the <A href="/wiki/Distributive_property" title="Distributive property">distributive property</A> of the dot product as follows:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle v^{2}=\mathbf {v} \cdot \mathbf {v} =(\mathbf {v} _{0}+\mathbf {a} t)\cdot (\mathbf {v} _{0}+\mathbf {a} t)=v_{0}^{2}+2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66ed09917363013c2ae3aa1d6898c489a3280bee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:57.847ex; height:3.343ex;" alt="v^{2}=\mathbf {v} \cdot \mathbf {v} =(\mathbf {v} _{0}+\mathbf {a} t)\cdot (\mathbf {v} _{0}+\mathbf {a} t)=v_{0}^{2}+2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}" /></SPAN></DD>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle (2\mathbf {a} )\cdot (\mathbf {r} -\mathbf {r} _{0})=(2\mathbf {a} )\cdot \left(\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\right)=2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}=v^{2}-v_{0}^{2}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30bc65adde716d0ed34cf90fb80f88bd3965e920" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:66.818ex; height:4.843ex;" alt="{\displaystyle (2\mathbf {a} )\cdot (\mathbf {r} -\mathbf {r} _{0})=(2\mathbf {a} )\cdot \left(\mathbf {v} _{0}t+{\tfrac {1}{2}}\mathbf {a} t^{2}\right)=2t(\mathbf {a} \cdot \mathbf {v} _{0})+a^{2}t^{2}=v^{2}-v_{0}^{2}}" /></SPAN></DD>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \therefore v^{2}=v_{0}^{2}+2(\mathbf {a} \cdot (\mathbf {r} -\mathbf {r} _{0}))}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3973ccac3423f889c90413b5e3a040361636d20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:26.535ex; height:3.343ex;" alt="\therefore v^{2}=v_{0}^{2}+2(\mathbf {a} \cdot (\mathbf {r} -\mathbf {r} _{0}))" /></SPAN></DD>
</DL>
<H4><SPAN class="mw-headline" id="Applications">Applications</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=7" title="Edit section: Applications">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<P>Elementary and frequent examples in kinematics involve <A href="/wiki/Projectile" title="Projectile">projectiles</A>, for example a ball thrown upwards into the air. Given initial speed <SPAN class="texhtml"><I>u</I></SPAN>, one can calculate how high the ball will travel before it begins to fall. The acceleration is local acceleration of gravity <SPAN class="texhtml"><I>g</I></SPAN>. At this point one must remember that while these quantities appear to be <A href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalars</A>, the direction of displacement, speed and acceleration is important. They could in fact be considered as unidirectional vectors. Choosing <SPAN class="texhtml"><I>s</I></SPAN> to measure up from the ground, the acceleration <SPAN class="texhtml"><I>a</I></SPAN> must be in fact <SPAN class="texhtml"><I>−g</I></SPAN>, since the force of <A href="/wiki/Gravity" title="Gravity">gravity</A> acts downwards and therefore also the acceleration on the ball due to it.</P>
<P>At the highest point, the ball will be at rest: therefore <SPAN class="texhtml"><I>v</I> = 0</SPAN>. Using equation [4] in the set above, we have:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>s</mi>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msup>
                <mi>v</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
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              <mo>−<!-- − --></mo>
              <msup>
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    <annotation encoding="application/x-tex">{\displaystyle s={\frac {v^{2}-u^{2}}{-2g}}.}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75248709be7324af38ad68fd15e6a2a01f44185" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:13.156ex; height:6.176ex;" alt="s={\frac {v^{2}-u^{2}}{-2g}}." /></SPAN></DD>
</DL>
<P>Substituting and cancelling minus signs gives:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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        <mi>s</mi>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msup>
              <mi>u</mi>
              <mrow class="MJX-TeXAtom-ORD">
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            </msup>
            <mrow>
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          </mfrac>
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        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle s={\frac {u^{2}}{2g}}.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6668a18959a94c2b45b2dad4ad9d966574bd0e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.338ex; width:8.105ex; height:6.176ex;" alt="s={\frac {u^{2}}{2g}}." /></SPAN></DD>
</DL>
<H4><SPAN class="mw-headline" id="Constant_circular_acceleration">Constant circular acceleration</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=8" title="Edit section: Constant circular acceleration">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H4>
<P>The analogues of the above equations can be written for <A href="/wiki/Rotation" title="Rotation">rotation</A>. Again these axial vectors must all be parallel to the axis of rotation, so only the magnitudes of the vectors are necessary,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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            <mtr>
              <mtd>
                <mi>ω<!-- ω --></mi>
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                  <mi>ω<!-- ω --></mi>
                  <mrow class="MJX-TeXAtom-ORD">
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            </mtr>
            <mtr>
              <mtd>
                <mi>θ<!-- θ --></mi>
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              <mtd>
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                  <mi>θ<!-- θ --></mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mi></mi>
                <mo>=</mo>
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                  <mi>θ<!-- θ --></mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mo>+</mo>
                <mrow class="MJX-TeXAtom-ORD">
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                <mo stretchy="false">(</mo>
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                  <mrow class="MJX-TeXAtom-ORD">
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                <mo>+</mo>
                <mi>ω<!-- ω --></mi>
                <mo stretchy="false">)</mo>
                <mi>t</mi>
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            <mtr>
              <mtd>
                <msup>
                  <mi>ω<!-- ω --></mi>
                  <mrow class="MJX-TeXAtom-ORD">
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                <mo stretchy="false">(</mo>
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                  <mi>θ<!-- θ --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0fbdf3867e338ce56138703c88a9877c62e31c2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -8.135ex; margin-bottom: -0.203ex; width:22.686ex; height:17.843ex;" alt="{\begin{aligned}\omega &=\omega _{0}+\alpha t\\\theta &=\theta _{0}+\omega _{0}t+{\tfrac {1}{2}}\alpha t^{2}\\\theta &=\theta _{0}+{\tfrac {1}{2}}(\omega _{0}+\omega )t\\\omega ^{2}&=\omega _{0}^{2}+2\alpha (\theta -\theta _{0})\\\theta &=\theta _{0}+\omega t-{\tfrac {1}{2}}\alpha t^{2}\\\end{aligned}}" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>α</I></SPAN> is the constant <A href="/wiki/Angular_acceleration" title="Angular acceleration">angular acceleration</A>, <SPAN class="texhtml"><I>ω</I></SPAN> is the <A href="/wiki/Angular_velocity" title="Angular velocity">angular velocity</A>, <SPAN class="texhtml"><I>ω</I><SUB>0</SUB></SPAN> is the initial angular velocity, <SPAN class="texhtml"><I>θ</I></SPAN> is the angle turned through (<A href="/wiki/Angular_displacement" title="Angular displacement">angular displacement</A>), <SPAN class="texhtml"><I>θ</I><SUB>0</SUB></SPAN> is the initial angle, and <SPAN class="texhtml"><I>t</I></SPAN> is the time taken to rotate from the initial state to the final state.</P>
<H3><SPAN class="mw-headline" id="General_planar_motion">General planar motion</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=9" title="Edit section: General planar motion">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV role="note" class="hatnote navigation-not-searchable">Main article: <A href="/wiki/Centripetal_force#General_planar_motion" title="Centripetal force">General planar motion</A></DIV>
<DIV class="thumb tmulti tright">
<DIV class="thumbinner" style="width:462px;max-width:462px">
<DIV class="tsingle" style="float:left;margin:1px;width:102px;max-width:102px">
<DIV class="thumbimage"><A href="/wiki/File:Position_vector_plane_polar_coords.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Position_vector_plane_polar_coords.svg/100px-Position_vector_plane_polar_coords.svg.png" width="100" height="137" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Position_vector_plane_polar_coords.svg/150px-Position_vector_plane_polar_coords.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Position_vector_plane_polar_coords.svg/200px-Position_vector_plane_polar_coords.svg.png 2x" data-file-width="155" data-file-height="212" /></A></DIV>
<DIV class="thumbcaption" style="clear:left">Position vector <SPAN class="texhtml"><B>r</B></SPAN>, always points radially from the origin.</DIV>
</DIV>
<DIV class="tsingle" style="float:left;margin:1px;width:152px;max-width:152px">
<DIV class="thumbimage"><A href="/wiki/File:Velocity_vector_plane_polar_coords.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Velocity_vector_plane_polar_coords.svg/150px-Velocity_vector_plane_polar_coords.svg.png" width="150" height="125" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Velocity_vector_plane_polar_coords.svg/225px-Velocity_vector_plane_polar_coords.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Velocity_vector_plane_polar_coords.svg/300px-Velocity_vector_plane_polar_coords.svg.png 2x" data-file-width="255" data-file-height="212" /></A></DIV>
<DIV class="thumbcaption" style="clear:left">Velocity vector <SPAN class="texhtml"><B>v</B></SPAN>, always tangent to the path of motion.</DIV>
</DIV>
<DIV class="tsingle" style="float:left;margin:1px;width:202px;max-width:202px">
<DIV class="thumbimage"><A href="/wiki/File:Acceleration_vector_plane_polar_coords.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Acceleration_vector_plane_polar_coords.svg/200px-Acceleration_vector_plane_polar_coords.svg.png" width="200" height="123" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Acceleration_vector_plane_polar_coords.svg/300px-Acceleration_vector_plane_polar_coords.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Acceleration_vector_plane_polar_coords.svg/400px-Acceleration_vector_plane_polar_coords.svg.png 2x" data-file-width="326" data-file-height="201" /></A></DIV>
<DIV class="thumbcaption" style="clear:left">Acceleration vector <SPAN class="texhtml"><B>a</B></SPAN>, not parallel to the radial motion but offset by the angular and Coriolis accelerations, nor tangent to the path but offset by the centripetal and radial accelerations.</DIV>
</DIV>
<DIV style="clear:left"></DIV>
<DIV class="thumbcaption" style="clear:left;text-align:left;background-color:transparent">Kinematic vectors in plane polar coordinates. Notice the setup is not restricted to 2D space, but a plane in any higher dimension.</DIV>
</DIV>
</DIV>
<P>These are the kinematic equations for a particle traversing a path in a plane, described by position <SPAN class="texhtml"><B>r</B> = <B>r</B>(<I>t</I>)</SPAN>.<SUP id="cite_ref-13" class="reference"><A href="#cite_note-13">[13]</A></SUP> They are simply the time derivatives of the position vector in plane <A href="/wiki/Polar_coordinates" class="mw-redirect" title="Polar coordinates">polar coordinates</A> using the definitions of physical quantities above for angular velocity <SPAN class="texhtml"><I>ω</I></SPAN> and angular acceleration <SPAN class="texhtml"><I>α</I></SPAN>. These are instantaneous quantities which change with time.</P>
<P>The position of the particle is</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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          <mi mathvariant="bold">r</mi>
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          <mo stretchy="false">)</mo>
          <mo>,</mo>
          <mi>θ<!-- θ --></mi>
          <mo stretchy="false">(</mo>
          <mi>t</mi>
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {r} \left(r(t),\theta (t)\right)=r\mathbf {\hat {e}} _{r}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5eb1998d2a457a860daeafba0fb989ec0fa12e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:22.612ex; height:2.843ex;" alt="{\displaystyle \mathbf {r} =\mathbf {r} \left(r(t),\theta (t)\right)=r\mathbf {\hat {e}} _{r}}" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><B>ê</B><SUB><I>r</I></SUB></SPAN> and <SPAN class="texhtml"><B>ê</B><SUB><I>θ</I></SUB></SPAN> are the <A href="/wiki/Polar_coordinate_system#Vector_calculus" title="Polar coordinate system">polar</A> unit vectors. Differentiating with respect to time gives the velocity</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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              <mover>
                <mi mathvariant="bold">e</mi>
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          <mrow class="MJX-TeXAtom-ORD">
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        <mo>+</mo>
        <mi>r</mi>
        <mi>ω<!-- ω --></mi>
        <msub>
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            <mi>θ<!-- θ --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58a65e1825c17315b2911025d51b419d6086c60d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:17.704ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} =\mathbf {\hat {e}} _{r}{\frac {dr}{dt}}+r\omega \mathbf {\hat {e}} _{\theta }}" /></SPAN></DD>
</DL>
<P>with radial component <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>dr</I></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>dt</I></SPAN></SPAN></SPAN> and an additional component <SPAN class="texhtml"><I>rω</I></SPAN> due to the rotation. Differentiating with respect to time again obtains the acceleration</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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          <mi>r</mi>
          <msup>
            <mi>ω<!-- ω --></mi>
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              <mn>2</mn>
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          <mo>)</mo>
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          <mrow class="MJX-TeXAtom-ORD">
            <mi>r</mi>
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        <mo>+</mo>
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          <mi>r</mi>
          <mi>α<!-- α --></mi>
          <mo>+</mo>
          <mn>2</mn>
          <mi>ω<!-- ω --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mfrac>
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                <mi>t</mi>
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          <mo>)</mo>
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          <mrow class="MJX-TeXAtom-ORD">
            <mi>θ<!-- θ --></mi>
          </mrow>
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\left({\frac {d^{2}r}{dt^{2}}}-r\omega ^{2}\right)\mathbf {\hat {e}} _{r}+\left(r\alpha +2\omega {\frac {dr}{dt}}\right)\mathbf {\hat {e}} _{\theta }}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14021f78d63e4a2d797cd03b7b704450afb59407" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:41.386ex; height:6.343ex;" alt="{\displaystyle \mathbf {a} =\left({\frac {d^{2}r}{dt^{2}}}-r\omega ^{2}\right)\mathbf {\hat {e}} _{r}+\left(r\alpha +2\omega {\frac {dr}{dt}}\right)\mathbf {\hat {e}} _{\theta }}" /></SPAN></DD>
</DL>
<P>which breaks into the radial acceleration <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>d</I><SUP>2</SUP><I>r</I></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>dt</I><SUP>2</SUP></SPAN></SPAN></SPAN>, <A href="/wiki/Centripetal_acceleration" class="mw-redirect" title="Centripetal acceleration">centripetal acceleration</A> <SPAN class="texhtml">–<I>rω</I><SUP>2</SUP></SPAN>, <A href="/wiki/Coriolis_acceleration" class="mw-redirect" title="Coriolis acceleration">Coriolis acceleration</A> <SPAN class="texhtml">2<I>ω</I><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>dr</I></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>dt</I></SPAN></SPAN></SPAN>, and angular acceleration <SPAN class="texhtml"><I>rα</I></SPAN>.</P>
<P>Special cases of motion described be these equations are summarized qualitatively in the table below. Two have already been discussed above, in the cases that either the radial components or the angular components are zero, and the non-zero component of motion describes uniform acceleration.</P>
<TABLE class="wikitable">
<TBODY><TR>
<TH scope="col" width="20%"><B>State of motion</B></TH>
<TH scope="col" width="20%">Constant <SPAN class="texhtml"><I>r</I></SPAN></TH>
<TH scope="col" width="20%"><SPAN class="texhtml"><I>r</I></SPAN> linear in <SPAN class="texhtml"><I>t</I></SPAN></TH>
<TH scope="col" width="20%"><SPAN class="texhtml"><I>r</I></SPAN> quadratic in <SPAN class="texhtml"><I>t</I></SPAN></TH>
<TH scope="col" width="20%"><I>r</I> non-linear in <I>t</I></TH>
</TR>
<TR>
<TH>Constant <SPAN class="texhtml"><I>θ</I></SPAN></TH>
<TD>Stationary</TD>
<TD>Uniform translation (constant translational velocity)</TD>
<TD>Uniform translational acceleration</TD>
<TD>Non-uniform translation</TD>
</TR>
<TR>
<TH><SPAN class="texhtml"><I>θ</I></SPAN> linear in <SPAN class="texhtml"><I>t</I></SPAN></TH>
<TD>Uniform angular motion in a circle (constant angular velocity)</TD>
<TD>Uniform angular motion in a spiral, constant radial velocity</TD>
<TD>Angular motion in a spiral, constant radial acceleration</TD>
<TD>Angular motion in a spiral, varying radial acceleration</TD>
</TR>
<TR>
<TH><SPAN class="texhtml"><I>θ</I></SPAN> quadratic in <SPAN class="texhtml"><I>t</I></SPAN></TH>
<TD>Uniform angular acceleration in a circle</TD>
<TD>Uniform angular acceleration in a spiral, constant radial velocity</TD>
<TD>Uniform angular acceleration in a spiral, constant radial acceleration</TD>
<TD>Uniform angular acceleration in a spiral, varying radial acceleration</TD>
</TR>
<TR>
<TH><SPAN class="texhtml"><I>θ</I></SPAN> non-linear in <SPAN class="texhtml"><I>t</I></SPAN></TH>
<TD>Non-uniform angular acceleration in a circle</TD>
<TD>Non-uniform angular acceleration in a spiral, constant radial velocity</TD>
<TD>Non-uniform angular acceleration in a spiral, constant radial acceleration</TD>
<TD>Non-uniform angular acceleration in a spiral, varying radial acceleration</TD>
</TR>
</TBODY></TABLE>
<H3><SPAN class="mw-headline" id="General_3D_motion">General 3D motion</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=10" title="Edit section: General 3D motion">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV role="note" class="hatnote navigation-not-searchable">Main article: <A href="/wiki/Spherical_coordinate_system" title="Spherical coordinate system">Spherical coordinate system</A></DIV>
<P>In 3D space, the equations in spherical coordinates <SPAN class="texhtml">(<I>r</I>, <I>θ</I>, <I>φ</I>)</SPAN> with corresponding unit vectors <SPAN class="texhtml"><B>ê</B><SUB><I>r</I></SUB></SPAN>, <SPAN class="texhtml"><B>ê</B><SUB><I>θ</I></SUB></SPAN> and <SPAN class="texhtml"><B>ê</B><SUB><I>φ</I></SUB></SPAN>, the position, velocity, and acceleration generalize respectively to</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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              <mtd>
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                  <mi mathvariant="bold">v</mi>
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                <mi></mi>
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                  <mrow class="MJX-TeXAtom-ORD">
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>r</mi>
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                <mo>+</mo>
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                </mrow>
                <msub>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mrow class="MJX-TeXAtom-ORD">
                      <mover>
                        <mi mathvariant="bold">e</mi>
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                  <mrow class="MJX-TeXAtom-ORD">
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                      <mi>φ<!-- φ --></mi>
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                      <mi>t</mi>
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                <mspace width="thinmathspace"></mspace>
                <mi>sin</mi>
                <mo><!--  --></mo>
                <mi>θ<!-- θ --></mi>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mrow class="MJX-TeXAtom-ORD">
                      <mover>
                        <mi mathvariant="bold">e</mi>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>φ<!-- φ --></mi>
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              <mtd>
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                  <mi mathvariant="bold">a</mi>
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                <mi></mi>
                <mo>=</mo>
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                  <mo>(</mo>
                  <mi>a</mi>
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                  <mi>r</mi>
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                      <mo>(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mfrac>
                          <mrow>
                            <mi>d</mi>
                            <mi>θ<!-- θ --></mi>
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                          <mrow>
                            <mi>d</mi>
                            <mi>t</mi>
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                        </mfrac>
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                      <mo>)</mo>
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                    <mrow class="MJX-TeXAtom-ORD">
                      <mn>2</mn>
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                  <mo>−<!-- − --></mo>
                  <mi>r</mi>
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                      <mo>(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
                        <mfrac>
                          <mrow>
                            <mi>d</mi>
                            <mi>φ<!-- φ --></mi>
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                            <mi>d</mi>
                            <mi>t</mi>
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                      <mo>)</mo>
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                    <mrow class="MJX-TeXAtom-ORD">
                      <mn>2</mn>
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                  </msup>
                  <msup>
                    <mi>sin</mi>
                    <mrow class="MJX-TeXAtom-ORD">
                      <mn>2</mn>
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                  <mo><!--  --></mo>
                  <mi>θ<!-- θ --></mi>
                  <mo>)</mo>
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                <msub>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mrow class="MJX-TeXAtom-ORD">
                      <mover>
                        <mi mathvariant="bold">e</mi>
                        <mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>r</mi>
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              <mtd></mtd>
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                <mi></mi>
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                  <mrow class="MJX-TeXAtom-ORD">
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                          <mi>d</mi>
                          <mrow class="MJX-TeXAtom-ORD">
                            <mn>2</mn>
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                        <mi>θ<!-- θ --></mi>
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                      <mrow>
                        <mi>d</mi>
                        <msup>
                          <mi>t</mi>
                          <mrow class="MJX-TeXAtom-ORD">
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                  <mi>v</mi>
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                        <mi>θ<!-- θ --></mi>
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                  <mi>r</mi>
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                      <mo>(</mo>
                      <mrow class="MJX-TeXAtom-ORD">
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                            <mi>d</mi>
                            <mi>φ<!-- φ --></mi>
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                            <mi>d</mi>
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                      <mo>)</mo>
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                    <mrow class="MJX-TeXAtom-ORD">
                      <mn>2</mn>
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                  <mi>sin</mi>
                  <mo><!--  --></mo>
                  <mi>θ<!-- θ --></mi>
                  <mi>cos</mi>
                  <mo><!--  --></mo>
                  <mi>θ<!-- θ --></mi>
                  <mo>)</mo>
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                  <mrow class="MJX-TeXAtom-ORD">
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                        <mi mathvariant="bold">e</mi>
                        <mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
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                  <mrow class="MJX-TeXAtom-ORD">
                    <mi>θ<!-- θ --></mi>
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              <mtd></mtd>
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                  <mi>r</mi>
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                          <mrow class="MJX-TeXAtom-ORD">
                            <mn>2</mn>
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                        <mi>φ<!-- φ --></mi>
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                  <mi>sin</mi>
                  <mo><!--  --></mo>
                  <mi>θ<!-- θ --></mi>
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                  <mi>v</mi>
                  <mspace width="thinmathspace"></mspace>
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                  <mi>sin</mi>
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                  <mspace width="thinmathspace"></mspace>
                  <mi>cos</mi>
                  <mo><!--  --></mo>
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                    <mrow class="MJX-TeXAtom-ORD">
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                    <mi>φ<!-- φ --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r\mathbf {\hat {e}} _{r}\\\mathbf {v} &=v\mathbf {\hat {e}} _{r}+r\,{\frac {d\theta }{dt}}\mathbf {\hat {e}} _{\theta }+r\,{\frac {d\varphi }{dt}}\,\sin \theta \mathbf {\hat {e}} _{\varphi }\\\mathbf {a} &=\left(a-r\left({\frac {d\theta }{dt}}\right)^{2}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin ^{2}\theta \right)\mathbf {\hat {e}} _{r}\\&+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2v{\frac {d\theta }{dt}}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin \theta \cos \theta \right)\mathbf {\hat {e}} _{\theta }\\&+\left(r{\frac {d^{2}\varphi }{dt^{2}}}\,\sin \theta +2v\,{\frac {d\varphi }{dt}}\,\sin \theta +2r\,{\frac {d\theta }{dt}}\,{\frac {d\varphi }{dt}}\,\cos \theta \right)\mathbf {\hat {e}} _{\varphi }\end{aligned}}\,\!}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1918de095b80ec9e8e43e002fa5ab9e5c6101a62" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -14.567ex; margin-right: -0.387ex; margin-bottom: -0.271ex; width:55.076ex; height:30.843ex;" alt="{\displaystyle {\begin{aligned}\mathbf {r} &=\mathbf {r} \left(t\right)=r\mathbf {\hat {e}} _{r}\\\mathbf {v} &=v\mathbf {\hat {e}} _{r}+r\,{\frac {d\theta }{dt}}\mathbf {\hat {e}} _{\theta }+r\,{\frac {d\varphi }{dt}}\,\sin \theta \mathbf {\hat {e}} _{\varphi }\\\mathbf {a} &=\left(a-r\left({\frac {d\theta }{dt}}\right)^{2}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin ^{2}\theta \right)\mathbf {\hat {e}} _{r}\\&+\left(r{\frac {d^{2}\theta }{dt^{2}}}+2v{\frac {d\theta }{dt}}-r\left({\frac {d\varphi }{dt}}\right)^{2}\sin \theta \cos \theta \right)\mathbf {\hat {e}} _{\theta }\\&+\left(r{\frac {d^{2}\varphi }{dt^{2}}}\,\sin \theta +2v\,{\frac {d\varphi }{dt}}\,\sin \theta +2r\,{\frac {d\theta }{dt}}\,{\frac {d\varphi }{dt}}\,\cos \theta \right)\mathbf {\hat {e}} _{\varphi }\end{aligned}}\,\!}" /></SPAN></DD>
</DL>
<P>In the case of a constant <SPAN class="texhtml"><I>φ</I></SPAN> this reduces to the planar equations above.</P>
<H2><SPAN class="mw-headline" id="Dynamic_equations_of_motion">Dynamic equations of motion</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=11" title="Edit section: Dynamic equations of motion">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<H3><SPAN class="mw-headline" id="Newtonian_mechanics">Newtonian mechanics</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=12" title="Edit section: Newtonian mechanics">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV role="note" class="hatnote navigation-not-searchable">Main article: <A href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</A></DIV>
<P>The first general equation of motion developed was <A href="/wiki/Newton%27s_second_law" class="mw-redirect" title="Newton's second law">Newton's second law</A> of motion, in its most general form states the rate of change of momentum <SPAN class="texhtml"><B>p</B> = <B>p</B>(<I>t</I>) = <I>m</I><B>v</B>(<I>t</I>)</SPAN> of an object equals the force <SPAN class="texhtml"><B>F</B> = <B>F</B>(<B>x</B>(<I>t</I>), <B>v</B>(<I>t</I>), <I>t</I>)</SPAN> acting on it,<SUP id="cite_ref-Mechanics.2C_D._Kleppner_2010.2C_p._112_14-0" class="reference"><A href="#cite_note-Mechanics.2C_D._Kleppner_2010.2C_p._112-14">[14]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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      <mstyle displaystyle="true" scriptlevel="0">
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          <mi mathvariant="bold">F</mi>
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</DL>
<P>The force in the equation is <I>not</I> the force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =m\mathbf {a} }</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c60fab89e8c3193952047dc565bcf8d233d115" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:8.163ex; height:2.176ex;" alt="\mathbf {F} =m\mathbf {a} " /></SPAN></DD>
</DL>
<P>since <SPAN class="texhtml"><I>m</I></SPAN> is a constant in <A href="/wiki/Newtonian_mechanics" class="mw-redirect" title="Newtonian mechanics">Newtonian mechanics</A>.</P>
<P>Newton's second law applies to point-like particles, and to all points in a <A href="/wiki/Rigid_body" title="Rigid body">rigid body</A>. They also apply to each point in a mass continua, like deformable solids or fluids, but the motion of the system must be accounted for, see <A href="/wiki/Material_derivative" title="Material derivative">material derivative</A>. In the case the mass is not constant, it is not sufficient to use the <A href="/wiki/Product_rule" title="Product rule">product rule</A> for the time derivative on the mass and velocity, and Newton's second law requires some modification consistent with <A href="/wiki/Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</A>, see <A href="/wiki/Variable-mass_system" title="Variable-mass system">variable-mass system</A>.</P>
<P>It may be simple to write down the equations of motion in vector form using Newton's laws of motion, but the components may vary in complicated ways with spatial coordinates and time, and solving them is not easy. Often there is an excess of variables to solve for the problem completely, so Newton's laws are not always the most efficient way to determine the motion of a system. In simple cases of rectangular geometry, Newton's laws work fine in Cartesian coordinates, but in other coordinate systems can become dramatically complex.</P>
<P>The momentum form is preferable since this is readily generalized to more complex systems, generalizes to <A href="/wiki/Special_relativity" title="Special relativity">special</A> and <A href="/wiki/General_relativity" title="General relativity">general relativity</A> (see <A href="/wiki/Four-momentum" title="Four-momentum">four-momentum</A>).<SUP id="cite_ref-Mechanics.2C_D._Kleppner_2010.2C_p._112_14-1" class="reference"><A href="#cite_note-Mechanics.2C_D._Kleppner_2010.2C_p._112-14">[14]</A></SUP> It can also be used with the momentum conservation. However, Newton's laws are not more fundamental than momentum conservation, because Newton's laws are merely consistent with the fact that zero resultant force acting on an object implies constant momentum, while a resultant force implies the momentum is not constant. Momentum conservation is always true for an isolated system not subject to resultant forces.</P>
<P>For a number of particles (see <A href="/wiki/Many_body_problem" class="mw-redirect" title="Many body problem">many body problem</A>), the equation of motion for one particle <SPAN class="texhtml"><I>i</I></SPAN> influenced by other particles is<SUP id="cite_ref-Relativity.2C_J.R._Forshaw_2009_8-1" class="reference"><A href="#cite_note-Relativity.2C_J.R._Forshaw_2009-8">[8]</A></SUP><SUP id="cite_ref-15" class="reference"><A href="#cite_note-15">[15]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {p} _{i}}{dt}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f35a2e849b6c0b74c9744cdfa9f35712cea09c93" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; margin-right: -0.387ex; width:20.838ex; height:7.176ex;" alt="{\frac {d\mathbf {p} _{i}}{dt}}=\mathbf {F} _{E}+\sum _{i\neq j}\mathbf {F} _{ij}\,\!" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><B>p</B><SUB><I>i</I></SUB></SPAN> is the momentum of particle <SPAN class="texhtml"><I>i</I></SPAN>, <SPAN class="texhtml"><B>F</B><SUB><I>ij</I></SUB></SPAN> is the force on particle <SPAN class="texhtml"><I>i</I></SPAN> by particle <SPAN class="texhtml"><I>j</I></SPAN>, and <SPAN class="texhtml"><B>F</B><SUB><I>E</I></SUB></SPAN> is the resultant external force due to any agent not part of system. Particle <SPAN class="texhtml"><I>i</I></SPAN> does not exert a force on itself.</P>
<P><A href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion">Euler's laws of motion</A> are similar to Newton's laws, but they are applied specifically to the motion of <A href="/wiki/Rigid_body" title="Rigid body">rigid bodies</A>. The <A href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</A> combine the forces and torques acting on a rigid body into a single equation.</P>
<P>Newton's second law for rotation takes a similar form to the translational case,<SUP id="cite_ref-16" class="reference"><A href="#cite_note-16">[16]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}\,,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ceccc0d28e9a72f7cdf92af6578da6e2bf7fa6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:9.263ex; height:5.509ex;" alt="{\boldsymbol {\tau }}={\frac {d\mathbf {L} }{dt}}\,," /></SPAN></DD>
</DL>
<P>by equating the <A href="/wiki/Torque" title="Torque">torque</A> acting on the body to the rate of change of its <A href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</A> <SPAN class="texhtml"><B>L</B></SPAN>. Analogous to mass times acceleration, the <A href="/wiki/Moment_of_inertia" title="Moment of inertia">moment of inertia</A> <A href="/wiki/Tensor" title="Tensor">tensor</A> <SPAN class="texhtml"><B>I</B></SPAN> depends on the distribution of mass about the axis of rotation, and the angular acceleration is the rate of change of angular velocity,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=\mathbf {I} \cdot {\boldsymbol {\alpha }}.}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c94e01edcf50947a4e5376e37d1939e35b1b12fd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.688ex; height:2.176ex;" alt="{\boldsymbol {\tau }}=\mathbf {I} \cdot {\boldsymbol {\alpha }}." /></SPAN></DD>
</DL>
<P>Again, these equations apply to point like particles, or at each point of a rigid body.</P>
<P>Likewise, for a number of particles, the equation of motion for one particle <SPAN class="texhtml"><I>i</I></SPAN> is<SUP id="cite_ref-17" class="reference"><A href="#cite_note-17">[17]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {L} _{i}}{dt}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5b2578bde3cfb814da84d81153a84997145715" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:20.676ex; height:7.009ex;" alt="{\frac {d\mathbf {L} _{i}}{dt}}={\boldsymbol {\tau }}_{E}+\sum _{i\neq j}{\boldsymbol {\tau }}_{ij}\,," /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><B>L</B><SUB><I>i</I></SUB></SPAN> is the angular momentum of particle <SPAN class="texhtml"><I>i</I></SPAN>, <SPAN class="texhtml"><B>τ</B><SUB><I>ij</I></SUB></SPAN> the torque on particle <SPAN class="texhtml"><I>i</I></SPAN> by particle <SPAN class="texhtml"><I>j</I></SPAN>, and <SPAN class="texhtml"><B>τ</B><SUB><I>E</I></SUB></SPAN> is resultant external torque (due to any agent not part of system). Particle <SPAN class="texhtml"><I>i</I></SPAN> does not exert a torque on itself.</P>
<H3><SPAN class="mw-headline" id="Applications_2">Applications</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=13" title="Edit section: Applications">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>Some examples<SUP id="cite_ref-Waves_1983_18-0" class="reference"><A href="#cite_note-Waves_1983-18">[18]</A></SUP> of Newton's law include describing the motion of a <A href="/wiki/Simple_pendulum" class="mw-redirect" title="Simple pendulum">simple pendulum</A>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle -mg\sin \theta =m{\frac {d^{2}(l\theta )}{dt^{2}}}\quad \Rightarrow \quad {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{l}}\sin \theta \,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc5f3bfb5e0ba420981b5f77fde6b543f8469958" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:46.872ex; height:6.176ex;" alt="{\displaystyle -mg\sin \theta =m{\frac {d^{2}(l\theta )}{dt^{2}}}\quad \Rightarrow \quad {\frac {d^{2}\theta }{dt^{2}}}=-{\frac {g}{l}}\sin \theta \,,}" /></SPAN></DD>
</DL>
<P>and a <A href="/wiki/Harmonic_oscillator#Sinusoidal_driving_force" title="Harmonic oscillator">damped, sinusoidally driven harmonic oscillator</A>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle F_{0}\sin(\omega t)=m\left({\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}x\right)\,.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0827bbd1669dd97511f12ddd8aef6a1dbe4cee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:42.623ex; height:6.343ex;" alt="F_{0}\sin(\omega t)=m\left({\frac {d^{2}x}{dt^{2}}}+2\zeta \omega _{0}{\frac {dx}{dt}}+\omega _{0}^{2}x\right)\,." /></SPAN></DD>
</DL>
<P>For describing the motion of masses due to gravity, <A href="/wiki/Newton%27s_law_of_gravity" class="mw-redirect" title="Newton's law of gravity">Newton's law of gravity</A> can be combined with Newton's second law. For two examples, a ball of mass <SPAN class="texhtml"><I>m</I></SPAN> thrown in the air, in air currents (such as wind) described by a vector field of resistive forces <SPAN class="texhtml"><B>R</B> = <B>R</B>(<B>r</B>, <I>t</I>)</SPAN>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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              <mi>m</mi>
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              <msup>
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    <annotation encoding="application/x-tex">{\displaystyle -{\frac {GmM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {R} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}+0\quad \Rightarrow \quad {\frac {d^{2}\mathbf {r} }{dt^{2}}}=-{\frac {GM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {A} \,\!}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1cfbbfcc9e76b2db4f40d25786e889e49dc3e6e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; margin-right: -0.387ex; width:59.769ex; height:7.009ex;" alt="-{\frac {GmM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {R} =m{\frac {d^{2}\mathbf {r} }{dt^{2}}}+0\quad \Rightarrow \quad {\frac {d^{2}\mathbf {r} }{dt^{2}}}=-{\frac {GM}{|\mathbf {r} |^{2}}}\mathbf {\hat {e}} _{r}+\mathbf {A} \,\!" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>G</I></SPAN> is the <A href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</A>, <SPAN class="texhtml"><I>M</I></SPAN> the mass of the Earth, and <SPAN class="texhtml"><B>A</B> = <SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><B>R</B></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>m</I></SPAN></SPAN></SPAN> is the acceleration of the projectile due to the air currents at position <SPAN class="texhtml"><B>r</B></SPAN> and time <SPAN class="texhtml"><I>t</I></SPAN>.</P>
<P>The classical <A href="/wiki/Many_body_problem" class="mw-redirect" title="Many body problem"><SPAN class="texhtml"><I>N</I></SPAN>-body problem</A> for <SPAN class="texhtml"><I>N</I></SPAN> particles each interacting with each other due to gravity is a set of <SPAN class="texhtml"><I>N</I></SPAN> nonlinear coupled second order ODEs,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}\mathbf {r} _{i}}{dt^{2}}}=G\sum _{i\neq j}{\frac {m_{i}m_{j}}{|\mathbf {r} _{j}-\mathbf {r} _{i}|^{3}}}(\mathbf {r} _{j}-\mathbf {r} _{i})}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a63828493c69fab3822816ab03180f673d069394" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:32.774ex; height:7.343ex;" alt="{\frac {d^{2}\mathbf {r} _{i}}{dt^{2}}}=G\sum _{i\neq j}{\frac {m_{i}m_{j}}{|\mathbf {r} _{j}-\mathbf {r} _{i}|^{3}}}(\mathbf {r} _{j}-\mathbf {r} _{i})" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>i</I> = 1, 2, …, <I>N</I></SPAN> labels the quantities (mass, position, etc.) associated with each particle.</P>
<H2><SPAN class="mw-headline" id="Analytical_mechanics">Analytical mechanics</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=14" title="Edit section: Analytical mechanics">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV role="note" class="hatnote navigation-not-searchable">Main articles: <A href="/wiki/Analytical_mechanics" title="Analytical mechanics">Analytical mechanics</A>, <A href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian mechanics</A> and <A href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</A></DIV>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:252px;"><A href="/wiki/File:Least_action_principle.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Least_action_principle.svg/250px-Least_action_principle.svg.png" width="250" height="182" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Least_action_principle.svg/375px-Least_action_principle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Least_action_principle.svg/500px-Least_action_principle.svg.png 2x" data-file-width="318" data-file-height="232" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Least_action_principle.svg" class="internal" title="Enlarge"></A></DIV>
As the system evolves, <SPAN class="texhtml"><B>q</B></SPAN> traces a path through <A href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</A> (only some are shown). The path taken by the system (red) has a stationary action (<SPAN class="texhtml"><I>δS</I> = 0</SPAN>) under small changes in the configuration of the system (<SPAN class="texhtml"><I>δ</I><B>q</B></SPAN>).<SUP id="cite_ref-19" class="reference"><A href="#cite_note-19">[19]</A></SUP></DIV>
</DIV>
</DIV>
<P>Using all three coordinates of 3D space is unnecessary if there are constraints on the system. If the system has <SPAN class="texhtml"><I>N</I></SPAN> <A href="/wiki/Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degrees of freedom</A>, then one can use a set of <SPAN class="texhtml"><I>N</I></SPAN> <A href="/wiki/Generalized_coordinates" title="Generalized coordinates">generalized coordinates</A> <SPAN class="texhtml"><B>q</B>(<I>t</I>) = [<I>q</I><SUB>1</SUB>(<I>t</I>), <I>q</I><SUB>2</SUB>(<I>t</I>) ... <I>q<SUB>N</SUB></I>(<I>t</I>)]</SPAN>, to define the configuration of the system. They can be in the form of <A href="/wiki/Arc_length" title="Arc length">arc lengths</A> or <A href="/wiki/Angle" title="Angle">angles</A>. They are a considerable simplification to describe motion, since they take advantage of the intrinsic constraints that limit the system's motion, and the number of coordinates is reduced to a minimum. The <A href="/wiki/Time_derivative" title="Time derivative">time derivatives</A> of the generalized coordinates are the <I>generalized velocities</I></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {\dot {q}} ={\frac {d\mathbf {q} }{dt}}\,.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8966bd8ea91d44376487b22da9170bcef666ee0d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:9.068ex; height:5.509ex;" alt="{\displaystyle \mathbf {\dot {q}} ={\frac {d\mathbf {q} }{dt}}\,.}" /></SPAN></DD>
</DL>
<P>The <A href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Euler–Lagrange equations</A> are<SUP id="cite_ref-Analytical_Mechanics_2008_2-1" class="reference"><A href="#cite_note-Analytical_Mechanics_2008-2">[2]</A></SUP><SUP id="cite_ref-Classical_Mechanics_1973_20-0" class="reference"><A href="#cite_note-Classical_Mechanics_1973-20">[20]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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        </mrow>
        <mrow>
          <mo>(</mo>
          <mrow class="MJX-TeXAtom-ORD">
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                <mi mathvariant="normal">∂<!-- ∂ --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5db7c9f259426f691e498a7ead4a563d04de53" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:18.411ex; height:6.176ex;" alt="{\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,," /></SPAN></DD>
</DL>
<P>where the <I>Lagrangian</I> is a function of the configuration <SPAN class="texhtml"><B>q</B></SPAN> and its time rate of change <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>d</I><B>q</B></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>dt</I></SPAN></SPAN></SPAN> (and possibly time <SPAN class="texhtml"><I>t</I></SPAN>)</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
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        <mi>L</mi>
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    <annotation encoding="application/x-tex">{\displaystyle L=L\left[\mathbf {q} (t),\mathbf {\dot {q}} (t),t\right]\,.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a30f80cbaa05dc7438902e9efb67fbad8ea096c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.58ex; height:2.843ex;" alt="L=L\left[\mathbf {q} (t),\mathbf {\dot {q}} (t),t\right]\,." /></SPAN></DD>
</DL>
<P>Setting up the Lagrangian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled <SPAN class="texhtml"><I>N</I></SPAN> second order <A href="/wiki/Ordinary_differential_equation" title="Ordinary differential equation">ODEs</A> in the coordinates are obtained.</P>
<P><A href="/wiki/Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamilton's equations</A> are<SUP id="cite_ref-Analytical_Mechanics_2008_2-2" class="reference"><A href="#cite_note-Analytical_Mechanics_2008-2">[2]</A></SUP><SUP id="cite_ref-Classical_Mechanics_1973_20-1" class="reference"><A href="#cite_note-Classical_Mechanics_1973-20">[20]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978b69872d211bfa764866383772528b71d36401" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:26.053ex; height:6.009ex;" alt="\mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,," /></SPAN></DD>
</DL>
<P>where the Hamiltonian</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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        <mi>H</mi>
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            <mi mathvariant="bold">q</mi>
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        <mspace width="thinmathspace"></mspace>
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    <annotation encoding="application/x-tex">{\displaystyle H=H\left[\mathbf {q} (t),\mathbf {p} (t),t\right]\,,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aecc975922b815e9bb266e1e3158865400b860c7" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:21.611ex; height:2.843ex;" alt="H=H\left[\mathbf {q} (t),\mathbf {p} (t),t\right]\,," /></SPAN></DD>
</DL>
<P>is a function of the configuration <SPAN class="texhtml"><B>q</B></SPAN> and conjugate <I><A href="/wiki/Canonical_coordinates" title="Canonical coordinates">"generalized" momenta</A></I></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}\,,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e54a6eb749972ccf587339d63c48b6a247b0c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:9.407ex; height:6.009ex;" alt="{\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}\,,}" /></SPAN></DD>
</DL>
<P>in which <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;">∂</SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">∂<B>q</B></SPAN></SPAN> = <BIG><BIG>(</BIG></BIG><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;">∂</SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">∂<I>q</I><SUB>1</SUB></SPAN></SPAN>, <SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;">∂</SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">∂<I>q</I><SUB>2</SUB></SPAN></SPAN>, …, <SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;">∂</SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;">∂<I>q</I><SUB><I>N</I></SUB></SPAN></SPAN><BIG><BIG>)</BIG></BIG></SPAN> is a shorthand notation for a vector of <A href="/wiki/Partial_derivative" title="Partial derivative">partial derivatives</A> with respect to the indicated variables (see for example <A href="/wiki/Matrix_calculus#Scope" title="Matrix calculus">matrix calculus</A> for this denominator notation), and possibly time <SPAN class="texhtml"><I>t</I></SPAN>,</P>
<P>Setting up the Hamiltonian of the system, then substituting into the equations and evaluating the partial derivatives and simplifying, a set of coupled <SPAN class="texhtml">2<I>N</I></SPAN> first order ODEs in the coordinates <SPAN class="texhtml"><I>q<SUB>i</SUB></I></SPAN> and momenta <SPAN class="texhtml"><I>p<SUB>i</SUB></I></SPAN> are obtained.</P>
<P>The <A href="/wiki/Hamilton%E2%80%93Jacobi_equation" title="Hamilton–Jacobi equation">Hamilton–Jacobi equation</A> is<SUP id="cite_ref-Analytical_Mechanics_2008_2-3" class="reference"><A href="#cite_note-Analytical_Mechanics_2008-2">[2]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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        <mo>−<!-- − --></mo>
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              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <mi>S</mi>
              <mo stretchy="false">(</mo>
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              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <mi>t</mi>
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        <mo>=</mo>
        <mi>H</mi>
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    <annotation encoding="application/x-tex">{\displaystyle -{\frac {\partial S(\mathbf {q} ,t)}{\partial t}}=H\left(\mathbf {q} ,\mathbf {p} ,t\right)\,.}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635b3978c29277746187ffc6a630448a72f0b74e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:25.336ex; height:5.843ex;" alt="-{\frac {\partial S(\mathbf {q} ,t)}{\partial t}}=H\left(\mathbf {q} ,\mathbf {p} ,t\right)\,." /></SPAN></DD>
</DL>
<P>where</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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        <mi>S</mi>
        <mo stretchy="false">[</mo>
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          <mi mathvariant="bold">q</mi>
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        <mo>,</mo>
        <mi>t</mi>
        <mo stretchy="false">]</mo>
        <mo>=</mo>
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          <mo>∫<!-- ∫ --></mo>
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        <mi>L</mi>
        <mo stretchy="false">(</mo>
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        <mo>,</mo>
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        <mi>t</mi>
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    <annotation encoding="application/x-tex">{\displaystyle S[\mathbf {q} ,t]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt\,,}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dde602b4562727a396922b4b680ebe4afdc9828" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:26.509ex; height:6.509ex;" alt="S[\mathbf {q} ,t]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)\,dt\,," /></SPAN></DD>
</DL>
<P>is <I>Hamilton's principal function</I>, also called the <I><A href="/wiki/Action_(physics)" title="Action (physics)">classical action</A></I> is a <A href="/wiki/Functional_(mathematics)" title="Functional (mathematics)">functional</A> of <SPAN class="texhtml"><I>L</I></SPAN>. In this case, the momenta are given by</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <mi>S</mi>
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            <mrow>
              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">q</mi>
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\frac {\partial S}{\partial \mathbf {q} }}\,.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26c5c35d460c66b33d4e641b176736d80e941b0c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:9.323ex; height:6.009ex;" alt="{\displaystyle \mathbf {p} ={\frac {\partial S}{\partial \mathbf {q} }}\,.}" /></SPAN></DD>
</DL>
<P>Although the equation has a simple general form, for a given Hamiltonian it is actually a single first order <I><A href="/wiki/Non-linear_differential_equation" class="mw-redirect" title="Non-linear differential equation">non-linear</A></I> <A href="/wiki/Partial_differential_equation" title="Partial differential equation">PDE</A>, in <SPAN class="texhtml"><I>N</I> + 1</SPAN> variables. The action <SPAN class="texhtml"><I>S</I></SPAN> allows identification of conserved quantities for mechanical systems, even when the mechanical problem itself cannot be solved fully, because any <A href="/wiki/Derivative" title="Derivative">differentiable</A> <A href="/wiki/Symmetry_in_physics" class="mw-redirect" title="Symmetry in physics">symmetry</A> of the <A href="/wiki/Action_(physics)" title="Action (physics)">action</A> of a physical system has a corresponding <A href="/wiki/Conservation_law" title="Conservation law">conservation law</A>, a theorem due to <A href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</A>.</P>
<P>All classical equations of motion can be derived from the <A href="/wiki/Variational_principle" title="Variational principle">variational principle</A> known as <A href="/wiki/Principle_of_least_action" title="Principle of least action">Hamilton's principle of least action</A></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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        <mi>δ<!-- δ --></mi>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \delta S=0\,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b262e01767c459369f9c9ed891d3449e289c9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.895ex; height:2.676ex;" alt="\delta S=0\,," /></SPAN></DD>
</DL>
<P>stating the path the system takes through the <A href="/wiki/Configuration_space_(physics)" title="Configuration space (physics)">configuration space</A> is the one with the least action <SPAN class="texhtml"><I>S</I></SPAN>.</P>
<H2><SPAN class="mw-headline" id="Electrodynamics">Electrodynamics</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=15" title="Edit section: Electrodynamics">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:202px;"><A href="/wiki/File:Lorentz_force_particle.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Lorentz_force_particle.svg/200px-Lorentz_force_particle.svg.png" width="200" height="212" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Lorentz_force_particle.svg/300px-Lorentz_force_particle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Lorentz_force_particle.svg/400px-Lorentz_force_particle.svg.png 2x" data-file-width="325" data-file-height="345" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Lorentz_force_particle.svg" class="internal" title="Enlarge"></A></DIV>
<A href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</A> <SPAN class="texhtml"><B>f</B></SPAN> on a <A href="/wiki/Charged_particle" title="Charged particle">charged particle</A> (of <A href="/wiki/Electric_charge" title="Electric charge">charge</A> <SPAN class="texhtml"><I>q</I></SPAN>) in motion (instantaneous velocity <SPAN class="texhtml"><B>v</B></SPAN>). The <A href="/wiki/Electric_field" title="Electric field"><SPAN class="texhtml"><B>E</B></SPAN> field</A> and <A href="/wiki/Magnetic_field" title="Magnetic field"><SPAN class="texhtml"><B>B</B></SPAN> field</A> vary in space and time.</DIV>
</DIV>
</DIV>
<P>In electrodynamics, the force on a charged particle of charge <SPAN class="texhtml"><I>q</I></SPAN> is the <A href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</A>:<SUP id="cite_ref-21" class="reference"><A href="#cite_note-21">[21]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">F</mi>
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        <mo>=</mo>
        <mi>q</mi>
        <mrow>
          <mo>(</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">E</mi>
          </mrow>
          <mo>+</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">v</mi>
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          <mo>×<!-- × --></mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">B</mi>
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\,\!}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34dd044922ed43cdc1f4d302457ea07ff264d6fa" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:19.289ex; height:2.843ex;" alt="\mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)\,\!" /></SPAN></DD>
</DL>
<P>Combining with Newton's second law gives a first order differential equation of motion, in terms of position of the particle:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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      <mstyle displaystyle="true" scriptlevel="0">
        <mi>m</mi>
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              <msup>
                <mi>d</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
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              </msup>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">r</mi>
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                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
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        <mo>=</mo>
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          <mo>(</mo>
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          </mrow>
          <mo>×<!-- × --></mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">B</mi>
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    <annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}\mathbf {r} }{dt^{2}}}=q\left(\mathbf {E} +{\frac {d\mathbf {r} }{dt}}\times \mathbf {B} \right)\,\!}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1173cada88e6b67e73cc175d7fc7b6286878f96a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:27.251ex; height:6.343ex;" alt="m{\frac {d^{2}\mathbf {r} }{dt^{2}}}=q\left(\mathbf {E} +{\frac {d\mathbf {r} }{dt}}\times \mathbf {B} \right)\,\!" /></SPAN></DD>
</DL>
<P>or its momentum:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
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              <mi>d</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">p</mi>
              </mrow>
            </mrow>
            <mrow>
              <mi>d</mi>
              <mi>t</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mi>q</mi>
        <mrow>
          <mo>(</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">E</mi>
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          <mo>+</mo>
          <mrow class="MJX-TeXAtom-ORD">
            <mfrac>
              <mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">p</mi>
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                <mo>×<!-- × --></mo>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">B</mi>
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              <mi>m</mi>
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          <mo>)</mo>
        </mrow>
        <mspace width="thinmathspace"></mspace>
        <mspace width="negativethinmathspace"></mspace>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {d\mathbf {p} }{dt}}=q\left(\mathbf {E} +{\frac {\mathbf {p} \times \mathbf {B} }{m}}\right)\,\!}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d33b916b166c413b9ae51c2dc78534a6cd06bf73" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; margin-right: -0.387ex; width:23.676ex; height:6.176ex;" alt="{\frac {d\mathbf {p} }{dt}}=q\left(\mathbf {E} +{\frac {\mathbf {p} \times \mathbf {B} }{m}}\right)\,\!" /></SPAN></DD>
</DL>
<P>The same equation can be obtained using the <A href="/wiki/Lagrangian_mechanics" title="Lagrangian mechanics">Lagrangian</A> (and applying Lagrange's equations above) for a charged particle of mass <SPAN class="texhtml"><I>m</I></SPAN> and charge <SPAN class="texhtml"><I>q</I></SPAN>:<SUP id="cite_ref-22" class="reference"><A href="#cite_note-22">[22]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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        <mi>L</mi>
        <mo>=</mo>
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          <mstyle displaystyle="false" scriptlevel="0">
            <mfrac>
              <mn>1</mn>
              <mn>2</mn>
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        </mrow>
        <mi>m</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mover>
              <mi mathvariant="bold">r</mi>
              <mo mathvariant="bold">˙<!-- ˙ --></mo>
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        <mo>⋅<!-- ⋅ --></mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mover>
              <mi mathvariant="bold">r</mi>
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        <mo>+</mo>
        <mi>q</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">A</mi>
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        <mo>⋅<!-- ⋅ --></mo>
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          <mrow class="MJX-TeXAtom-ORD">
            <mover>
              <mrow class="MJX-TeXAtom-ORD">
                <mi mathvariant="bold">r</mi>
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              <mo>˙<!-- ˙ --></mo>
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        <mo>−<!-- − --></mo>
        <mi>q</mi>
        <mi>ϕ<!-- ϕ --></mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle L={\tfrac {1}{2}}m\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot {\dot {\mathbf {r} }}-q\phi }</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066808135c69b860292e150d0d770b4ebf73d72d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.171ex; width:26.952ex; height:3.509ex;" alt="{\displaystyle L={\tfrac {1}{2}}m\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot {\dot {\mathbf {r} }}-q\phi }" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><B>A</B></SPAN> and <SPAN class="texhtml"><I>ϕ</I></SPAN> are the electromagnetic <A href="/wiki/Electrical_potential" class="mw-redirect" title="Electrical potential">scalar</A> and <A href="/wiki/Magnetic_potential" title="Magnetic potential">vector</A> potential fields. The Lagrangian indicates an additional detail: the <A href="/wiki/Canonical_momentum" class="mw-redirect" title="Canonical momentum">canonical momentum</A> in Lagrangian mechanics is given by:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} }</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a01904ef0fbdd3aee925656403b2c571fccc464" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:20.997ex; height:5.509ex;" alt="\mathbf {P} ={\frac {\partial L}{\partial {\dot {\mathbf {r} }}}}=m{\dot {\mathbf {r} }}+q\mathbf {A} " /></SPAN></DD>
</DL>
<P>instead of just <SPAN class="texhtml"><I>m</I><B>v</B></SPAN>, implying the motion of a charged particle is fundamentally determined by the mass and charge of the particle. The Lagrangian expression was first used to derive the force equation.</P>
<P>Alternatively the Hamiltonian (and substituting into the equations):<SUP id="cite_ref-Classical_Mechanics_1973_20-2" class="reference"><A href="#cite_note-Classical_Mechanics_1973-20">[20]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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                  <mi mathvariant="bold">A</mi>
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        <mo>+</mo>
        <mi>q</mi>
        <mi>ϕ<!-- ϕ --></mi>
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    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle H={\frac {\left(\mathbf {P} -q\mathbf {A} \right)^{2}}{2m}}+q\phi \,\!}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301d001cfd95bd6bca4df903ca21fc058e9aad35" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; margin-right: -0.387ex; width:22.423ex; height:6.176ex;" alt="H={\frac {\left(\mathbf {P} -q\mathbf {A} \right)^{2}}{2m}}+q\phi \,\!" /></SPAN></DD>
</DL>
<P>can derive the Lorentz force equation.</P>
<H2><SPAN class="mw-headline" id="General_relativity">General relativity</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=16" title="Edit section: General relativity">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<H3><SPAN class="mw-headline" id="Geodesic_equation_of_motion">Geodesic equation of motion</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=17" title="Edit section: Geodesic equation of motion">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:202px;"><A href="/wiki/File:Geodesic_deviation_on_a_sphere.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Geodesic_deviation_on_a_sphere.svg/200px-Geodesic_deviation_on_a_sphere.svg.png" width="200" height="223" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Geodesic_deviation_on_a_sphere.svg/300px-Geodesic_deviation_on_a_sphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Geodesic_deviation_on_a_sphere.svg/400px-Geodesic_deviation_on_a_sphere.svg.png 2x" data-file-width="220" data-file-height="245" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Geodesic_deviation_on_a_sphere.svg" class="internal" title="Enlarge"></A></DIV>
Geodesics on a <A href="/wiki/Sphere" title="Sphere">sphere</A> are arcs of <A href="/wiki/Great_circle" title="Great circle">great circles</A> (yellow curve). On a <A href="/wiki/Two-dimensional_space" title="Two-dimensional space">2D</A>–<A href="/wiki/Manifold" title="Manifold">manifold</A> (such as the sphere shown), the direction of the accelerating geodesic is uniquely fixed if the separation vector <SPAN class="texhtml"><B>ξ</B></SPAN> is <A href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</A> to the "fiducial geodesic" (green curve). As the separation vector <SPAN class="texhtml"><B>ξ</B><SUB>0</SUB></SPAN> changes to <SPAN class="texhtml"><B>ξ</B></SPAN> after a distance <SPAN class="texhtml"><I>s</I></SPAN>, the geodesics are not parallel (geodesic deviation).<SUP id="cite_ref-23" class="reference"><A href="#cite_note-23">[23]</A></SUP></DIV>
</DIV>
</DIV>
<DIV role="note" class="hatnote navigation-not-searchable">Main articles: <A href="/wiki/Geodesics_in_general_relativity" title="Geodesics in general relativity">Geodesics in general relativity</A> and <A href="/wiki/Geodesic_equation" class="mw-redirect" title="Geodesic equation">Geodesic equation</A></DIV>
<P>The above equations are valid in flat spacetime. In <A href="/wiki/Curved_space" title="Curved space">curved space</A> <A href="/wiki/Spacetime" title="Spacetime">spacetime</A>, things become mathematically more complicated since there is no straight line; this is generalized and replaced by a <I><A href="/wiki/Geodesic" title="Geodesic">geodesic</A></I> of the curved spacetime (the shortest length of curve between two points). For curved <A href="/wiki/Manifold" title="Manifold">manifolds</A> with a <A href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</A> <SPAN class="texhtml"><I>g</I></SPAN>, the metric provides the notion of arc length (see <A href="/wiki/Line_element" title="Line element">line element</A> for details), the <A href="/wiki/Differential_(infinitesimal)" title="Differential (infinitesimal)">differential</A> arc length is given by:<SUP id="cite_ref-24" class="reference"><A href="#cite_note-24">[24]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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                <mi>α<!-- α --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle ds={\sqrt {g_{\alpha \beta }dx^{\alpha }dx^{\beta }}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d586a45bfdfa6c22d1825df0c6be4eb85f383b3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.671ex; width:18.737ex; height:4.843ex;" alt="ds={\sqrt {g_{\alpha \beta }dx^{\alpha }dx^{\beta }}}" /></SPAN></DD>
</DL>
<P>and the geodesic equation is a second-order differential equation in the coordinates, the general solution is a family of geodesics:<SUP id="cite_ref-25" class="reference"><A href="#cite_note-25">[25]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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                  <mi>μ<!-- μ --></mi>
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          <mrow class="MJX-TeXAtom-ORD">
            <mi>α<!-- α --></mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{2}x^{\mu }}{ds^{2}}}=-\Gamma ^{\mu }{}_{\alpha \beta }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a57a4520d040276d56b096d122af89a77bdb8fd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:24.838ex; height:6.009ex;" alt="{\frac {d^{2}x^{\mu }}{ds^{2}}}=-\Gamma ^{\mu }{}_{\alpha \beta }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>Γ <SUP>μ</SUP><SUB>αβ</SUB></I></SPAN> is a <A href="/wiki/Christoffel_symbols#Christoffel_symbols_of_the_second_kind_.28symmetric_definition.29" title="Christoffel symbols">Christoffel symbol of the second kind</A>, which contains the metric (with respect to the coordinate system).</P>
<P>Given the <A href="/wiki/Mass-energy_equivalence" class="mw-redirect" title="Mass-energy equivalence">mass-energy</A> distribution provided by the <A href="/wiki/Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</A> <SPAN class="texhtml"><I>T <SUP>αβ</SUP></I></SPAN>, the <A href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</A> are a set of non-linear second-order partial differential equations in the metric, and imply the curvature of space time is equivalent to a gravitational field (see <A href="/wiki/Principle_of_equivalence" class="mw-redirect" title="Principle of equivalence">principle of equivalence</A>). Mass falling in curved spacetime is equivalent to a mass falling in a gravitational field - because <A href="/wiki/Fictitious_force#Gravity_as_a_fictitious_force" title="Fictitious force">gravity is a fictitious force</A>. The <I>relative acceleration</I> of one geodesic to another in curved spacetime is given by the <I><A href="/wiki/Geodesic_deviation_equation" class="mw-redirect" title="Geodesic deviation equation">geodesic deviation equation</A></I>:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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                  <mn>2</mn>
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              </msup>
              <msup>
                <mi>ξ<!-- ξ --></mi>
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        <mo>=</mo>
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          <mi>R</mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {D^{2}\xi ^{\alpha }}{ds^{2}}}=-R^{\alpha }{}_{\beta \gamma \delta }{\frac {dx^{\alpha }}{ds}}\xi ^{\gamma }{\frac {dx^{\delta }}{ds}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea6a3cc5f0c288beb4dce94cda1782276a23e15" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:28.246ex; height:6.009ex;" alt="{\frac {D^{2}\xi ^{\alpha }}{ds^{2}}}=-R^{\alpha }{}_{\beta \gamma \delta }{\frac {dx^{\alpha }}{ds}}\xi ^{\gamma }{\frac {dx^{\delta }}{ds}}" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><B>ξ</B><SUP><I>α</I></SUP> = <I>x</I><SUB>2</SUB><SUP><I>α</I></SUP> − <I>x</I><SUB>1</SUB><SUP><I>α</I></SUP></SPAN> is the separation vector between two geodesics, <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>D</I></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>ds</I></SPAN></SPAN></SPAN> (<I>not</I> just <SPAN class="texhtml"><SPAN class="sfrac nowrap" style="display:inline-block; vertical-align:-0.5em; font-size:85%; text-align:center;"><SPAN style="display:block; line-height:1em; margin:0 0.1em;"><I>d</I></SPAN><SPAN class="visualhide">/</SPAN><SPAN style="display:block; line-height:1em; margin:0 0.1em; border-top:1px solid;"><I>ds</I></SPAN></SPAN></SPAN>) is the <A href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivative</A>, and <SPAN class="texhtml"><I>R<SUP>α</SUP><SUB>βγδ</SUB></I></SPAN> is the <A href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</A>, containing the Christoffel symbols. In other words, the geodesic deviation equation is the equation of motion for masses in curved spacetime, analogous to the Lorentz force equation for charges in an electromagnetic field.<SUP id="cite_ref-26" class="reference"><A href="#cite_note-26">[26]</A></SUP></P>
<P>For flat spacetime, the metric is a constant tensor so the Christoffel symbols vanish, and the geodesic equation has the solutions of straight lines. This is also the limiting case when masses move according to <A href="/wiki/Newton%27s_law_of_gravity" class="mw-redirect" title="Newton's law of gravity">Newton's law of gravity</A>.</P>
<H3><SPAN class="mw-headline" id="Spinning_objects">Spinning objects</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=18" title="Edit section: Spinning objects">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>In general relativity, rotational motion is described by the <A href="/wiki/Relativistic_angular_momentum" title="Relativistic angular momentum">relativistic angular momentum</A> tensor, including the <A href="/wiki/Spin_tensor" title="Spin tensor">spin tensor</A>, which enter the equations of motion under <A href="/wiki/Covariant_derivative" title="Covariant derivative">covariant derivatives</A> with respect to <A href="/wiki/Proper_time" title="Proper time">proper time</A>. The <A href="/wiki/Mathisson%E2%80%93Papapetrou%E2%80%93Dixon_equations" title="Mathisson–Papapetrou–Dixon equations">Mathisson–Papapetrou–Dixon equations</A> describe the motion of spinning objects moving in a <A href="/wiki/Gravitational_field" title="Gravitational field">gravitational field</A>.</P>
<H2><SPAN class="mw-headline" id="Analogues_for_waves_and_fields">Analogues for waves and fields</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=19" title="Edit section: Analogues for waves and fields">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Unlike the equations of motion for describing particle mechanics, which are systems of coupled ordinary differential equations, the analogous equations governing the dynamics of <A href="/wiki/Wave_(physics)" class="mw-redirect" title="Wave (physics)">waves</A> and <A href="/wiki/Field_(physics)" title="Field (physics)">fields</A> are always <A href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</A>, since the waves or fields are functions of space and time. For a particular solution, <A href="/wiki/Boundary_conditions" class="mw-redirect" title="Boundary conditions">boundary conditions</A> along with initial conditions need to be specified.</P>
<P>Sometimes in the following contexts, the wave or field equations are also called "equations of motion".</P>
<H3><SPAN class="mw-headline" id="Field_equations">Field equations</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=20" title="Edit section: Field equations">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>Equations that describe the spatial dependence and <A href="/wiki/Time_evolution" title="Time evolution">time evolution</A> of fields are called <I><A href="/wiki/Field_equation" title="Field equation">field equations</A></I>. These include</P>
<UL>
<LI><A href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</A> for the <A href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</A>,</LI>
<LI><A href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</A> for <A href="/wiki/Newtonian_gravitation" class="mw-redirect" title="Newtonian gravitation">Newtonian gravitational</A> or <A href="/wiki/Electrostatic" class="mw-redirect" title="Electrostatic">electrostatic</A> field potentials,</LI>
<LI>the <A href="/wiki/Einstein_field_equation" class="mw-redirect" title="Einstein field equation">Einstein field equation</A> for <A href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitation</A> (<A href="/wiki/Newton%27s_law_of_gravity" class="mw-redirect" title="Newton's law of gravity">Newton's law of gravity</A> is a special case for weak gravitational fields and low velocities of particles).</LI>
</UL>
<P>This terminology is not universal: for example although the <A href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</A> govern the <A href="/wiki/Velocity_field" class="mw-redirect" title="Velocity field">velocity field</A> of a <A href="/wiki/Fluid" title="Fluid">fluid</A>, they are not usually called "field equations", since in this context they represent the momentum of the fluid and are called the "momentum equations" instead.</P>
<H3><SPAN class="mw-headline" id="Wave_equations">Wave equations</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=21" title="Edit section: Wave equations">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>Equations of wave motion are called <I><A href="/wiki/Wave_equation" title="Wave equation">wave equations</A></I>. The solutions to a wave equation give the time-evolution and spatial dependence of the <A href="/wiki/Amplitude" title="Amplitude">amplitude</A>. Boundary conditions determine if the solutions describe <A href="/wiki/Traveling_wave" class="mw-redirect" title="Traveling wave">traveling waves</A> or <A href="/wiki/Standing_waves" class="mw-redirect" title="Standing waves">standing waves</A>.</P>
<P>From classical equations of motion and field equations; mechanical, <A href="/wiki/Gravitational_wave" title="Gravitational wave">gravitational wave</A>, and <A href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</A> equations can be derived. The general linear wave equation in 3D is:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mn>1</mn>
            <msup>
              <mi>v</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msup>
                <mi mathvariant="normal">∂<!-- ∂ --></mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
              <mi>X</mi>
            </mrow>
            <mrow>
              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <msup>
                <mi>t</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <msup>
          <mi mathvariant="normal">∇<!-- ∇ --></mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>2</mn>
          </mrow>
        </msup>
        <mi>X</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{v^{2}}}{\frac {\partial ^{2}X}{\partial t^{2}}}=\nabla ^{2}X}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/550b8e0d5020f1353109376de439467b2850698f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:16.385ex; height:6.009ex;" alt="{\frac {1}{v^{2}}}{\frac {\partial ^{2}X}{\partial t^{2}}}=\nabla ^{2}X" /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>X</I> = <I>X</I>(<B>r</B>, <I>t</I>)</SPAN> is any mechanical or electromagnetic field amplitude, say:<SUP id="cite_ref-27" class="reference"><A href="#cite_note-27">[27]</A></SUP></P>
<UL>
<LI>the <A href="/wiki/Transverse_wave" title="Transverse wave">transverse</A> or <A href="/wiki/Longitudinal_wave" title="Longitudinal wave">longitudinal</A> <A href="/wiki/Displacement_(vector)" title="Displacement (vector)">displacement</A> of a vibrating rod, wire, cable, membrane etc.,</LI>
<LI>the fluctuating <A href="/wiki/Pressure" title="Pressure">pressure</A> of a medium, <A href="/wiki/Sound_pressure" title="Sound pressure">sound pressure</A>,</LI>
<LI>the <A href="/wiki/Electric_field" title="Electric field">electric fields</A> <SPAN class="texhtml"><B>E</B></SPAN> or <SPAN class="texhtml"><B>D</B></SPAN>, or the <A href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</A> <SPAN class="texhtml"><B>B</B></SPAN> or <SPAN class="texhtml"><B>H</B></SPAN>,</LI>
<LI>the <A href="/wiki/Voltage" title="Voltage">voltage</A> <SPAN class="texhtml"><I>V</I></SPAN> or <A href="/wiki/Electric_current" title="Electric current">current</A> <SPAN class="texhtml"><I>I</I></SPAN> in an <A href="/wiki/Alternating_current" title="Alternating current">alternating current</A> circuit,</LI>
</UL>
<P>and <SPAN class="texhtml"><I>v</I></SPAN> is the <A href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</A>. Nonlinear equations model the dependence of phase velocity on amplitude, replacing <SPAN class="texhtml"><I>v</I></SPAN> by <SPAN class="texhtml"><I>v</I>(<I>X</I>)</SPAN>. There are other linear and nonlinear wave equations for very specific applications, see for example the <A href="/wiki/Korteweg%E2%80%93de_Vries_equation" title="Korteweg–de Vries equation">Korteweg–de Vries equation</A>.</P>
<H3><SPAN class="mw-headline" id="Quantum_theory">Quantum theory</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=22" title="Edit section: Quantum theory">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>In quantum theory, the wave and field concepts both appear.</P>
<P>In <A href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</A>, in which particles also have wave-like properties according to <A href="/wiki/Wave%E2%80%93particle_duality" title="Wave–particle duality">wave–particle duality</A>, the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the <A href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</A> in its most general form:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>i</mi>
        <mi class="MJX-variant">ℏ<!-- ℏ --></mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <mi mathvariant="normal">Ψ<!-- Ψ --></mi>
            </mrow>
            <mrow>
              <mi mathvariant="normal">∂<!-- ∂ --></mi>
              <mi>t</mi>
            </mrow>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mover>
              <mi>H</mi>
              <mo stretchy="false">^<!-- ^ --></mo>
            </mover>
          </mrow>
        </mrow>
        <mi mathvariant="normal">Ψ<!-- Ψ --></mi>
        <mspace width="thinmathspace"></mspace>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,,}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e4912007a42006294bf98ea9c8ea84a225ce4d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.005ex; width:14.159ex; height:5.509ex;" alt="i\hbar {\frac {\partial \Psi }{\partial t}}={\hat {H}}\Psi \,," /></SPAN></DD>
</DL>
<P>where <SPAN class="texhtml"><I>Ψ</I></SPAN> is the <A href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</A> of the system, <SPAN class="texhtml"><I>Ĥ</I></SPAN> is the quantum <A href="/wiki/Hamiltonian_operator" class="mw-redirect" title="Hamiltonian operator">Hamiltonian operator</A>, rather than a function as in classical mechanics, and <SPAN class="texhtml"><I>ħ</I></SPAN> is the <A href="/wiki/Planck_constant" title="Planck constant">Planck constant</A> divided by 2<SPAN class="texhtml">π</SPAN>. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the <A href="/wiki/Correspondence_principle" title="Correspondence principle">correspondence principle</A>, in the limit that <SPAN class="texhtml"><I>ħ</I></SPAN> becomes zero.</P>
<P>Throughout all aspects of quantum theory, relativistic or non-relativistic, there are <A href="/wiki/Mathematical_formulation_of_quantum_mechanics" title="Mathematical formulation of quantum mechanics">various formulations</A> alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance:</P>
<UL>
<LI>the <A href="/wiki/Heisenberg_picture" title="Heisenberg picture">Heisenberg equation of motion</A> resembles the time evolution of classical observables as functions of position, momentum, and time, if one replaces dynamical observables by their <A href="/wiki/Operator_(physics)" title="Operator (physics)">quantum operators</A> and the classical <A href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</A> by the <A href="/wiki/Commutator" title="Commutator">commutator</A>,</LI>
<LI>the <A href="/wiki/Phase_space_formulation" title="Phase space formulation">phase space formulation</A> closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing,</LI>
<LI>the Feynman <A href="/wiki/Path_integral_formulation" title="Path integral formulation">path integral formulation</A> extends the <A href="/wiki/Principle_of_least_action" title="Principle of least action">principle of least action</A> to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.</LI>
</UL>
<H2><SPAN class="mw-headline" id="See_also">See also</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=23" title="Edit section: See also">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="div-col columns column-count column-count-3" style="-moz-column-count: 3; -webkit-column-count: 3; column-count: 3;">
<UL>
<LI><A href="/wiki/Scalar_(physics)" title="Scalar (physics)">Scalar (physics)</A></LI>
<LI><A href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">Vector</A></LI>
<LI><A href="/wiki/Distance" title="Distance">Distance</A></LI>
<LI><A href="/wiki/Displacement_(vector)" title="Displacement (vector)">Displacement</A></LI>
<LI><A href="/wiki/Speed" title="Speed">Speed</A></LI>
<LI><A href="/wiki/Velocity" title="Velocity">Velocity</A></LI>
<LI><A href="/wiki/Acceleration" title="Acceleration">Acceleration</A></LI>
<LI><A href="/wiki/Angular_displacement" title="Angular displacement">Angular displacement</A></LI>
<LI><A href="/wiki/Angular_speed" class="mw-redirect" title="Angular speed">Angular speed</A></LI>
<LI><A href="/wiki/Angular_velocity" title="Angular velocity">Angular velocity</A></LI>
<LI><A href="/wiki/Angular_acceleration" title="Angular acceleration">Angular acceleration</A></LI>
<LI><A href="/wiki/Equations_for_a_falling_body" title="Equations for a falling body">Equations for a falling body</A></LI>
<LI><A href="/wiki/Trajectory#Example:_Constant_gravity.2C_no_drag_or_wind" title="Trajectory">Parabolic trajectory</A></LI>
<LI><A href="/wiki/Curvilinear_coordinates" title="Curvilinear coordinates">Curvilinear coordinates</A></LI>
<LI><A href="/wiki/Orthogonal_coordinates" title="Orthogonal coordinates">Orthogonal coordinates</A></LI>
<LI><A href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</A></LI>
<LI><A href="/wiki/Torricelli%27s_equation" title="Torricelli's equation">Torricelli's equation</A></LI>
<LI><A href="/wiki/Euler%E2%80%93Lagrange_equation" title="Euler–Lagrange equation">Euler–Lagrange equation</A></LI>
<LI><A href="/wiki/Generalized_forces" title="Generalized forces">Generalized forces</A></LI>
<LI><A href="/wiki/Defining_equation_(physics)" title="Defining equation (physics)">Defining equation (physics)</A></LI>
<LI><A href="/wiki/Euler%27s_laws_of_motion" title="Euler's laws of motion">Newton–Euler laws of motion for a rigid body</A></LI>
</UL>
</DIV>
<H2><SPAN class="mw-headline" id="References">References</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Equations_of_motion&action=edit&section=24" title="Edit section: References">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="reflist" style="list-style-type: decimal;">
<DIV class="mw-references-wrap mw-references-columns">
<OL class="references">
<LI id="cite_note-Physics_1991-1"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Physics_1991_1-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><I>Encyclopaedia of Physics</I> (second Edition), R.G. Lerner, G.L. Trigg, VHC Publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1 (VHC Inc.) 0-89573-752-3</SPAN></LI>
<LI id="cite_note-Analytical_Mechanics_2008-2"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Analytical_Mechanics_2008_2-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Analytical_Mechanics_2008_2-1"><SUP><I><B>b</B></I></SUP></A> <A href="#cite_ref-Analytical_Mechanics_2008_2-2"><SUP><I><B>c</B></I></SUP></A> <A href="#cite_ref-Analytical_Mechanics_2008_2-3"><SUP><I><B>d</B></I></SUP></A></SPAN> <SPAN class="reference-text"><I>Analytical Mechanics</I>, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/978-0-521-57572-0" title="Special:BookSources/978-0-521-57572-0">978-0-521-57572-0</A></SPAN></LI>
<LI id="cite_note-3"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-3"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See <A href="/wiki/History_of_mathematics" title="History of mathematics">History of Mathematics</A></SPAN></LI>
<LI id="cite_note-book-4"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-book_4-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The Britannica Guide to History of Mathematics, ed. Erik Gregersen</SPAN></LI>
<LI id="cite_note-5"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-5"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Discourses, Galileo</SPAN></LI>
<LI id="cite_note-6"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-6"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Dialogues Concerning Two New Sciences, by Galileo Galilei; translated by Henry Crew, Alfonso De Salvio</SPAN></LI>
<LI id="cite_note-Halliday2004-7"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Halliday2004_7-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">Halliday, David; Resnick, Robert; Walker, Jearl (2004-06-16). <I>Fundamentals of Physics</I> (7 Sub ed.). Wiley. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-471-23231-9" title="Special:BookSources/0-471-23231-9">0-471-23231-9</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Physics&rft.edition=7+Sub&rft.pub=Wiley&rft.date=2004-06-16&rft.isbn=0-471-23231-9&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rft.au=Walker%2C+Jearl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-Relativity.2C_J.R._Forshaw_2009-8"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Relativity.2C_J.R._Forshaw_2009_8-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Relativity.2C_J.R._Forshaw_2009_8-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/978-0-470-01460-8" title="Special:BookSources/978-0-470-01460-8">978-0-470-01460-8</A></SPAN></LI>
<LI id="cite_note-9"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-9"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">M.R. Spiegel; S. Lipschutz; D. Spellman (2009). <I>Vector Analysis</I>. Schaum's Outlines (2nd ed.). McGraw Hill. p. 33. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/978-0-07-161545-7" title="Special:BookSources/978-0-07-161545-7">978-0-07-161545-7</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Analysis&rft.series=Schaum%27s+Outlines&rft.pages=33&rft.edition=2nd&rft.pub=McGraw+Hill&rft.date=2009&rft.isbn=978-0-07-161545-7&rft.au=M.R.+Spiegel&rft.au=S.+Lipschutz&rft.au=D.+Spellman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-Physics_P.M-10"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Physics_P.M_10-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Physics_P.M_10-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">Essential Principles of Physics, P.M. Whelan, M.J. Hodgeson, second Edition, 1978, John Murray, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-7195-3382-1" title="Special:BookSources/0-7195-3382-1">0-7195-3382-1</A></SPAN></LI>
<LI id="cite_note-Hanrahan2003-11"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Hanrahan2003_11-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">Hanrahan, Val; Porkess, R (2003). <I>Additional Mathematics for OCR</I>. London: Hodder & Stoughton. p. 219. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-340-86960-7" title="Special:BookSources/0-340-86960-7">0-340-86960-7</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Additional+Mathematics+for+OCR&rft.place=London&rft.pages=219&rft.pub=Hodder+%26+Stoughton&rft.date=2003&rft.isbn=0-340-86960-7&rft.aulast=Hanrahan&rft.aufirst=Val&rft.au=Porkess%2C+R&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-12"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-12"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">Keith Johnson (2001). <A rel="nofollow" class="external text" href="https://books.google.com/books?id=D4nrQDzq1jkC&pg=PA135&dq=suvat#v=onepage&q=suvat&f=false"><I>Physics for you: revised national curriculum edition for GCSE</I></A> (4th ed.). Nelson Thornes. p. 135. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/978-0-7487-6236-1" title="Special:BookSources/978-0-7487-6236-1">978-0-7487-6236-1</A>. <Q>The 5 symbols are remembered by "suvat". Given any three, the other two can be found.</Q></CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+you%3A+revised+national+curriculum+edition+for+GCSE&rft.pages=135&rft.edition=4th&rft.pub=Nelson+Thornes&rft.date=2001&rft.isbn=978-0-7487-6236-1&rft.au=Keith+Johnson&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DD4nrQDzq1jkC%26pg%3DPA135%26dq%3Dsuvat%23v%3Donepage%26q%3Dsuvat%26f%3Dfalse&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-13"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-13"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">3000 Solved Problems in Physics, Schaum Series, A. Halpern, Mc Graw Hill, 1988, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/978-0-07-025734-4" title="Special:BookSources/978-0-07-025734-4">978-0-07-025734-4</A></SPAN></LI>
<LI id="cite_note-Mechanics.2C_D._Kleppner_2010.2C_p._112-14"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Mechanics.2C_D._Kleppner_2010.2C_p._112_14-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Mechanics.2C_D._Kleppner_2010.2C_p._112_14-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">An Introduction to Mechanics, D. Kleppner, R.J. Kolenkow, Cambridge University Press, 2010, p. 112, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/978-0-521-19821-9" title="Special:BookSources/978-0-521-19821-9">978-0-521-19821-9</A></SPAN></LI>
<LI id="cite_note-15"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-15"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Encyclopaedia of Physics (second Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (VHC Inc.) 0-89573-752-3</SPAN></LI>
<LI id="cite_note-16"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-16"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">"Mechanics, D. Kleppner 2010"</SPAN></LI>
<LI id="cite_note-17"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-17"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">"Relativity, J.R. Forshaw 2009"</SPAN></LI>
<LI id="cite_note-Waves_1983-18"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Waves_1983_18-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The Physics of Vibrations and Waves (3rd edition), H.J. Pain, John Wiley & Sons, 1983, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-471-90182-2" title="Special:BookSources/0-471-90182-2">0-471-90182-2</A></SPAN></LI>
<LI id="cite_note-19"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-19"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">R. Penrose (2007). <I><A href="/wiki/The_Road_to_Reality" title="The Road to Reality">The Road to Reality</A></I>. Vintage books. p. 474. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-679-77631-1" title="Special:BookSources/0-679-77631-1">0-679-77631-1</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Road+to+Reality&rft.pages=474&rft.pub=Vintage+books&rft.date=2007&rft.isbn=0-679-77631-1&rft.au=R.+Penrose&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-Classical_Mechanics_1973-20"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Classical_Mechanics_1973_20-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Classical_Mechanics_1973_20-1"><SUP><I><B>b</B></I></SUP></A> <A href="#cite_ref-Classical_Mechanics_1973_20-2"><SUP><I><B>c</B></I></SUP></A></SPAN> <SPAN class="reference-text">Classical Mechanics (second edition), T.W.B. Kibble, European Physics Series, 1973, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-07-084018-0" title="Special:BookSources/0-07-084018-0">0-07-084018-0</A></SPAN></LI>
<LI id="cite_note-21"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-21"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Electromagnetism (second edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-471-92712-0" title="Special:BookSources/0-471-92712-0">0-471-92712-0</A></SPAN></LI>
<LI id="cite_note-22"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-22"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Classical Mechanics (second Edition), T.W.B. Kibble, European Physics Series, Mc Graw Hill (UK), 1973, <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-07-084018-0" title="Special:BookSources/0-07-084018-0">0-07-084018-0</A>.</SPAN></LI>
<LI id="cite_note-23"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-23"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Misner, Thorne, Wheeler, <I><A href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">Gravitation</A></I></SPAN></LI>
<LI id="cite_note-24"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-24"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">C.B. Parker (1994). <I>McGraw Hill Encyclopaedia of Physics</I> (second ed.). p. 1199. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-07-051400-3" title="Special:BookSources/0-07-051400-3">0-07-051400-3</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=McGraw+Hill+Encyclopaedia+of+Physics&rft.pages=1199&rft.edition=second&rft.date=1994&rft.isbn=0-07-051400-3&rft.au=C.B.+Parker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-25"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-25"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">C.B. Parker (1994). <I>McGraw Hill Encyclopaedia of Physics</I> (second ed.). p. 1200. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-07-051400-3" title="Special:BookSources/0-07-051400-3">0-07-051400-3</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=McGraw+Hill+Encyclopaedia+of+Physics&rft.pages=1200&rft.edition=second&rft.date=1994&rft.isbn=0-07-051400-3&rft.au=C.B.+Parker&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-26"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-26"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">J.A. Wheeler; C. Misner; K.S. Thorne (1973). <I>Gravitation</I>. W.H. Freeman & Co. pp. 34–35. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-7167-0344-0" title="Special:BookSources/0-7167-0344-0">0-7167-0344-0</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.pages=34-35&rft.pub=W.H.+Freeman+%26+Co&rft.date=1973&rft.isbn=0-7167-0344-0&rft.au=J.A.+Wheeler&rft.au=C.+Misner&rft.au=K.S.+Thorne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
<LI id="cite_note-27"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-27"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">H.D. Young; R.A. Freedman (2008). <I>University Physics</I> (12th ed.). Addison-Wesley (Pearson International). <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-321-50130-6" title="Special:BookSources/0-321-50130-6">0-321-50130-6</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=University+Physics&rft.edition=12th&rft.pub=Addison-Wesley+%28Pearson+International%29&rft.date=2008&rft.isbn=0-321-50130-6&rft.au=H.D.+Young&rft.au=R.A.+Freedman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEquations+of+motion" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN></SPAN></LI>
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document_title: Equations of motion

token: Equations

token: of

token: motion

token: wikipedia

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question_text: equation for the velocity of an object in motion at constant acceleration