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<title>Two-body problem</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Two-body problem</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about the two-body problem in classical mechanics. For the relativistic version, see <a href="Two-body_problem_in_general_relativity" title="Two-body problem in general relativity">Two-body problem in general relativity</a>. For the career management problem of working couples, see <a href="Two-body_problem_(career)" title="Two-body problem (career)">Two-body problem (career)</a>.</div>

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:308px;max-width:308px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"></span></div></div><div class="tsingle" style="width:102px;max-width:102px"><div class="thumbimage"><span typeof="mw:File"></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><b>Left:</b> Two bodies of similar <a href="Mass" title="Mass">mass</a> orbiting a common <a href="Barycenter" class="mw-redirect" title="Barycenter">barycenter</a> external to both bodies, with <a href="Elliptic_orbit" title="Elliptic orbit">elliptic orbits</a>. This model is typical of <a href="Binary_stars" class="mw-redirect" title="Binary stars">binary stars</a>.<br><b>Right:</b> Two bodies with a "slight" difference in mass orbiting a common barycenter. Their sizes and this type of orbit are similar to the <a href="Pluto#Satellites" title="Pluto">Pluto–Charon system</a> (in which the barycenter is external to both bodies), as well as the <a href="Earth" title="Earth">Earth</a>–<a href="Moon" title="Moon">Moon</a> system (in which the barycenter is internal to the larger body).</div></div></div></div>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of a series on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Astrodynamics" class="mw-redirect" title="Astrodynamics">Astrodynamics</a></th></tr><tr><td class="sidebar-image" style="padding-bottom:0.85em;"><span typeof="mw:File"></span></td></tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;">

<div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Orbital_mechanics" title="Orbital mechanics"><span style="font-size:110%;">Orbital mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="Orbital_elements" title="Orbital elements">Orbital elements</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Apsis" title="Apsis">Apsis</a></li>

<li><a href="Argument_of_periapsis" title="Argument of periapsis">Argument of periapsis</a></li>

<li><a href="Orbital_eccentricity" title="Orbital eccentricity">Eccentricity</a></li>

<li><a href="Orbital_inclination" title="Orbital inclination">Inclination</a></li>

<li><a href="Mean_anomaly" title="Mean anomaly">Mean anomaly</a></li>

<li><a href="Orbital_node" title="Orbital node">Orbital nodes</a></li>

<li><a href="Semi-major_and_semi-minor_axes" title="Semi-major and semi-minor axes">Semi-major axis</a></li>

<li><a href="True_anomaly" title="True anomaly">True anomaly</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Types of by <br>eccentricity</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Circular_orbit" title="Circular orbit">Circular orbit</a></li>

<li><a href="Elliptic_orbit" title="Elliptic orbit">Elliptic orbit</a></li></ul>

<div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Transfer_orbit" title="Transfer orbit">Transfer orbit</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="Hohmann_transfer_orbit" title="Hohmann transfer orbit">Hohmann transfer orbit</a></li><li><a href="Bi-elliptic_transfer" title="Bi-elliptic transfer">Bi-elliptic transfer orbit</a>)</li></ul></div></div>

<ul><li><a href="Parabolic_trajectory" title="Parabolic trajectory">Parabolic orbit</a></li>

<li><a href="Hyperbolic_trajectory" title="Hyperbolic trajectory">Hyperbolic orbit</a></li>

<li><a href="Radial_trajectory" title="Radial trajectory">Radial orbit</a></li>

<li><a href="Orbital_decay" title="Orbital decay">Decaying orbit</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Equations</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Dynamical_friction" title="Dynamical friction">Dynamical friction</a></li>

<li><a href="Escape_velocity" title="Escape velocity">Escape velocity</a></li>

<li><a href="Kepler's_equation" title="Kepler's equation">Kepler's equation</a></li>

<li><a href="Kepler's_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</a></li>

<li><a href="Orbital_period" title="Orbital period">Orbital period</a></li>

<li><a href="Orbital_speed" title="Orbital speed">Orbital velocity</a></li>

<li><a href="Surface_gravity" title="Surface gravity">Surface gravity</a></li>

<li><a href="Specific_orbital_energy" title="Specific orbital energy">Specific orbital energy</a></li>

<li><a href="Vis-viva_equation" title="Vis-viva equation">Vis-viva equation</a></li></ul></div></div></td>

</tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;">

<div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Celestial_mechanics" title="Celestial mechanics"><span style="font-size:110%;">Celestial mechanics</span></a></div></th></tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Gravitational influences</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Barycenter" class="mw-redirect" title="Barycenter">Barycenter</a></li>

<li><a href="Hill_sphere" title="Hill sphere">Hill sphere</a></li>

<li><a href="Perturbation_(astronomy)" title="Perturbation (astronomy)">Perturbations</a></li>

<li><a href="Sphere_of_influence_(astrodynamics)" title="Sphere of influence (astrodynamics)">Sphere of influence</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="N-body_problem" title="N-body problem">N-body orbits</a></div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Lagrange_point" title="Lagrange point">Lagrangian points</a> <div class="hlist" style="font-size:90%"><ul><li>(<a href="Halo_orbit" title="Halo orbit">Halo orbits</a>)</li></ul></div></div>

<ul><li><a href="Lissajous_orbit" title="Lissajous orbit">Lissajous orbits</a></li>

<li><a href="Lyapunov_stability" title="Lyapunov stability">Lyapunov orbits</a></li></ul></div></div></td>

</tr><tr><th class="sidebar-heading" style="padding-bottom:0.55em;">

<div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="Aerospace_engineering" title="Aerospace engineering"><span style="font-size:110%;">Engineering and efficiency</span></a></div></th></tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Preflight engineering</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Mass_ratio" title="Mass ratio">Mass ratio</a></li>

<li><a href="Payload_fraction" title="Payload fraction">Payload fraction</a></li>

<li><a href="Propellant_mass_fraction" title="Propellant mass fraction">Propellant mass fraction</a></li>

<li><a href="Tsiolkovsky_rocket_equation" title="Tsiolkovsky rocket equation">Tsiolkovsky rocket equation</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Efficiency measures</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Gravity_assist" title="Gravity assist">Gravity assist</a></li>

<li><a href="Oberth_effect" title="Oberth effect">Oberth effect</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Propulsive maneuvers</div><div class="sidebar-list-content mw-collapsible-content plainlist" style="padding-top:0;">

<ul><li><a href="Orbital_maneuver" title="Orbital maneuver">Orbital maneuver</a></li>

<li><a href="Orbit_insertion" title="Orbit insertion">Orbit insertion</a></li></ul></div></div></td>

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<p>In <a href="Classical_mechanics" title="Classical mechanics">classical mechanics</a>, the <b>two-body problem</b> is to calculate and predict the motion of two massive bodies that are orbiting each other in space. The problem assumes that the two bodies are <a href="Point_particle" title="Point particle">point particles</a> that interact only with one another; the only force affecting each object arises from the other one, and all other objects are ignored.

</p><p>The most prominent example of the classical two-body problem is the gravitational case (see also <a href="Kepler_problem" title="Kepler problem">Kepler problem</a>), arising in astronomy for predicting the orbits (or escapes from orbit) of objects such as <a href="Satellite" title="Satellite">satellites</a>, <a href="Planet" title="Planet">planets</a>, and <a href="Stars" class="mw-redirect" title="Stars">stars</a>. A two-point-particle model of such a system nearly always describes its behavior well enough to provide useful insights and predictions.

</p><p>A simpler "one body" model, the "<a href="Classical_central-force_problem" title="Classical central-force problem">central-force problem</a>", treats one object as the immobile source of a force acting on the other. One then seeks to predict the motion of the single remaining mobile object. Such an approximation can give useful results when one object is much more massive than the other (as with a light planet orbiting a heavy star, where the star can be treated as essentially stationary).

</p><p>However, the one-body approximation is usually unnecessary except as a stepping stone. For many forces, including gravitational ones, the general version of the two-body problem can be <a href="#Reduction_to_two_independent,_one-body_problems">reduced to a pair of one-body problems</a>, allowing it to be solved completely, and giving a solution simple enough to be used effectively.

</p><p>By contrast, the <a href="Three-body_problem" title="Three-body problem">three-body problem</a> (and, more generally, the <a href="N-body_problem" title="N-body problem"><i>n</i>-body problem</a> for <i>n</i> ≥ 3) cannot be solved in terms of first integrals, except in special cases.

</p>

<div class="mw-heading mw-heading2"><h2 id="Results_for_prominent_cases">Results for prominent cases</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Gravitation_and_other_inverse-square_examples">Gravitation and other inverse-square examples</h3></div>

<p>The two-body problem is interesting in astronomy because pairs of astronomical objects are often moving rapidly in arbitrary directions (so their motions become interesting), widely separated from one another (so they will not collide) and even more widely separated from other objects (so outside influences will be small enough to be ignored safely).

</p><p>Under the force of <a href="Gravity" title="Gravity">gravity</a>, each member of a pair of such objects will orbit their mutual center of mass in an elliptical pattern, unless they are moving fast enough to escape one another entirely, in which case their paths will diverge along other planar <a href="Conic_section" title="Conic section">conic sections</a>. If one object is very much heavier than the other, it will move far less than the other with reference to the shared center of mass. The mutual center of mass may even be inside the larger object.

</p><p>For the derivation of the solutions to the problem, see <a href="Classical_central-force_problem" title="Classical central-force problem">Classical central-force problem</a> or <a href="Kepler_problem" title="Kepler problem">Kepler problem</a>.

</p><p>In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive <a href="Scalar_potential" title="Scalar potential">scalar force field</a> obeying an <a href="Inverse-square_law" title="Inverse-square law">inverse-square law</a>, with <a href="Coulomb's_law" title="Coulomb's law">electrostatic attraction</a> being the obvious physical example. In practice, such problems rarely arise. Except perhaps in experimental apparatus or other specialized equipment, we rarely encounter electrostatically interacting objects which are moving fast enough, and in such a direction, as to avoid colliding, and/or which are isolated enough from their surroundings.

</p><p>The <a href="Dynamical_system" title="Dynamical system">dynamical system</a> of a two-body system under the influence of torque turns out to be a <a href="Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm-Liouville equation</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Inapplicability_to_atoms_and_subatomic_particles">Inapplicability to atoms and subatomic particles</h3></div>

<p>Although the two-body model treats the objects as point particles, classical mechanics only apply to systems of macroscopic scale. Most behavior of subatomic particles <i>cannot</i> be predicted under the classical assumptions underlying this article or using the mathematics here.

</p><p><a href="Electron" title="Electron">Electrons</a> in an atom are sometimes described as "orbiting" its <a href="Atomic_nucleus" title="Atomic nucleus">nucleus</a>, following an <a href="Bohr_model" title="Bohr model">early conjecture</a> of <a href="Niels_Bohr" title="Niels Bohr">Niels Bohr</a> (this is the source of the term "<a href="Atomic_orbital" title="Atomic orbital">orbital</a>"). However, electrons don't actually orbit nuclei in any meaningful sense, and <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> are necessary for any useful understanding of the electron's real behavior. Solving the classical two-body problem for an electron orbiting an atomic nucleus is misleading and does not produce many useful insights.

</p>

<div class="mw-heading mw-heading2"><h2 id="Reduction_to_two_independent,_one-body_problems">Reduction to two independent, one-body problems</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Classical_central-force_problem#Relation_to_the_classical_two-body_problem" title="Classical central-force problem">Classical central-force problem § Relation to the classical two-body problem</a></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Kepler_problem" title="Kepler problem">Kepler problem</a></div>

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<p>The complete two-body problem can be solved by re-formulating it as two one-body problems: a trivial one and one that involves solving for the motion of one particle in an external <a href="Potential" title="Potential">potential</a>. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved.

</p>

<p>Let <span class="texhtml"><b>x</b><sub>1</sub></span> and <span class="texhtml"><b>x</b><sub>2</sub></span> be the vector positions of the two bodies, and <i>m</i><sub>1</sub> and <i>m</i><sub>2</sub> be their masses. The goal is to determine the trajectories <span class="texhtml"><b>x</b><sub>1</sub>(<i>t</i>)</span> and <span class="texhtml"><b>x</b><sub>2</sub>(<i>t</i>)</span> for all times <i>t</i>, given the initial positions <span class="texhtml"><b>x</b><sub>1</sub>(<i>t</i> = 0)</span> and <span class="texhtml"><b>x</b><sub>2</sub>(<i>t</i> = 0)</span> and the initial velocities <span class="texhtml"><b>v</b><sub>1</sub>(<i>t</i> = 0)</span> and <span class="texhtml"><b>v</b><sub>2</sub>(<i>t</i> = 0)</span>.

</p><p>When applied to the two masses, <a href="Newton's_laws_of_motion#Newton's_second_law" title="Newton's laws of motion">Newton's second law</a> states that

</p>

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</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{1}{\ddot {\mathbf {x} }}_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{1}{\ddot {\mathbf {x} }}_{1}}</annotation>

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</math></span></span></td> <td></td> <td class="nowrap">Equation <span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{21}(\mathbf {x} _{1},\mathbf {x} _{2})=m_{2}{\ddot {\mathbf {x} }}_{2}}</annotation>

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</math></span></span></td> <td></td> <td class="nowrap">Equation <span id="math_2" class="reference nourlexpansion" style="font-weight:bold;">2</span></td></tr></tbody></table>

<p>where <b>F</b><sub>12</sub> is the force on mass 1 due to its interactions with mass 2, and <b>F</b><sub>21</sub> is the force on mass 2 due to its interactions with mass 1. The two dots on top of the <b>x</b> position vectors denote their second derivative with respect to time, or their acceleration vectors.

</p><p>Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. <i>Adding</i> equations (1) and (<b><a href="#math_2">2</a></b>) results in an equation describing the <a href="Center_of_mass" title="Center of mass">center of mass</a> (<a href="Barycenter" class="mw-redirect" title="Barycenter">barycenter</a>) motion. By contrast, <i>subtracting</i> equation (2) from equation (1) results in an equation that describes how the vector <span class="texhtml"><b>r</b> = <b>x</b><sub>1</sub> − <b>x</b><sub>2</sub></span> between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories <span class="texhtml"><b>x</b><sub>1</sub>(<i>t</i>)</span> and <span class="texhtml"><b>x</b><sub>2</sub>(<i>t</i>)</span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Center_of_mass_motion_(1st_one-body_problem)">Center of mass motion (1st one-body problem)</h3></div>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {R} }</annotation>

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</math></span><img src="./5de85fcc2a00d8ba14aae84aeef812d7fef4b3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.003ex; height:2.176ex;" alt="{\displaystyle \mathbf {R} }" loading="lazy"></span> be the position of the <a href="Center_of_mass" title="Center of mass">center of mass</a> (<a href="Barycenter" class="mw-redirect" title="Barycenter">barycenter</a>) of the system. Addition of the force equations (1) and (2) yields

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<annotation encoding="application/x-tex">{\displaystyle m_{1}{\ddot {\mathbf {x} }}_{1}+m_{2}{\ddot {\mathbf {x} }}_{2}=(m_{1}+m_{2}){\ddot {\mathbf {R} }}=\mathbf {F} _{12}+\mathbf {F} _{21}=0}</annotation>

</semantics>

</math></span></span>

where we have used <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's third law</a> <span class="texhtml"><b>F</b><sub>12</sub> = −<b>F</b><sub>21</sub></span> and where

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<annotation encoding="application/x-tex">{\displaystyle {\ddot {\mathbf {R} }}\equiv {\frac {m_{1}{\ddot {\mathbf {x} }}_{1}+m_{2}{\ddot {\mathbf {x} }}_{2}}{m_{1}+m_{2}}}.}</annotation>

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</math></span></span>

</p><p>The resulting equation:

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<annotation encoding="application/x-tex">{\displaystyle {\ddot {\mathbf {R} }}=0}</annotation>

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shows that the velocity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ={\frac {dR}{dt}}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\frac {dR}{dt}}}</annotation>

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</math></span><img src="./170d37ee8f4d0f3ea8e1eef60591731c14613648.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.325ex; height:5.509ex;" alt="{\displaystyle \mathbf {v} ={\frac {dR}{dt}}}" loading="lazy"></span> of the center of mass is constant, from which follows that the total momentum <span class="texhtml"><i>m</i><sub>1</sub> <b>v</b><sub>1</sub> + <i>m</i><sub>2</sub> <b>v</b><sub>2</sub></span> is also constant (<a href="Conservation_of_momentum" class="mw-redirect" title="Conservation of momentum">conservation of momentum</a>). Hence, the position <span class="texhtml"><b>R</b>(<i>t</i>)</span> of the center of mass can be determined at all times from the initial positions and velocities.

</p>

<div class="mw-heading mw-heading3"><h3 id="Displacement_vector_motion_(2nd_one-body_problem)">Displacement vector motion (2nd one-body problem)</h3></div>

<p>Dividing both force equations by the respective masses, subtracting the second equation from the first, and rearranging gives the equation

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<annotation encoding="application/x-tex">{\displaystyle {\ddot {\mathbf {r} }}={\ddot {\mathbf {x} }}_{1}-{\ddot {\mathbf {x} }}_{2}=\left({\frac {\mathbf {F} _{12}}{m_{1}}}-{\frac {\mathbf {F} _{21}}{m_{2}}}\right)=\left({\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}\right)\mathbf {F} _{12}}</annotation>

</semantics>

</math></span></span>

where we have again used <a href="Newton's_third_law" class="mw-redirect" title="Newton's third law">Newton's third law</a> <span class="texhtml"><b>F</b><sub>12</sub> = −<b>F</b><sub>21</sub></span> and where <span class="texhtml"><b>r</b></span> is the <a href="Displacement_(vector)" class="mw-redirect" title="Displacement (vector)">displacement vector</a> from mass 2 to mass 1, as defined above.

</p><p>The force between the two objects, which originates in the two objects, should only be a function of their separation <span class="texhtml"><b>r</b></span> and not of their absolute positions <span class="texhtml"><b>x</b><sub>1</sub></span> and <span class="texhtml"><b>x</b><sub>2</sub></span>; otherwise, there would not be <a href="Translational_symmetry" title="Translational symmetry">translational symmetry</a>, and the laws of physics would have to change from place to place. The subtracted equation can therefore be written:

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<annotation encoding="application/x-tex">{\displaystyle \mu {\ddot {\mathbf {r} }}=\mathbf {F} _{12}(\mathbf {x} _{1},\mathbf {x} _{2})=\mathbf {F} (\mathbf {r} )}</annotation>

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</math></span></span>

where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math></span><img src="./9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"></span> is the <b><a href="Reduced_mass" title="Reduced mass">reduced mass</a></b>

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<annotation encoding="application/x-tex">{\displaystyle \mu ={\frac {1}{{\frac {1}{m_{1}}}+{\frac {1}{m_{2}}}}}={\frac {m_{1}m_{2}}{m_{1}+m_{2}}}.}</annotation>

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</p><p>Solving the equation for <span class="texhtml"><b>r</b>(<i>t</i>)</span> is the key to the two-body problem. The solution depends on the specific force between the bodies, which is defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\mathbf {r} )}</annotation>

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</math></span><img src="./0b5f64c13605eb065971b62adf62deb92c2c9354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.594ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} (\mathbf {r} )}" loading="lazy"></span>. For the case where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (\mathbf {r} )}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\mathbf {r} )}</annotation>

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</math></span><img src="./0b5f64c13605eb065971b62adf62deb92c2c9354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.594ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} (\mathbf {r} )}" loading="lazy"></span> follows an <a href="Inverse-square_law" title="Inverse-square law">inverse-square law</a>, see the <a href="Kepler_problem" title="Kepler problem">Kepler problem</a>.

</p><p>Once <span class="texhtml"><b>R</b>(<i>t</i>)</span> and <span class="texhtml"><b>r</b>(<i>t</i>)</span> have been determined, the original trajectories may be obtained

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1}(t)=\mathbf {R} (t)+{\frac {m_{2}}{m_{1}+m_{2}}}\mathbf {r} (t)}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{2}(t)=\mathbf {R} (t)-{\frac {m_{1}}{m_{1}+m_{2}}}\mathbf {r} (t)}</annotation>

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as may be verified by substituting the definitions of <b>R</b> and <b>r</b> into the right-hand sides of these two equations.

</p>

<div class="mw-heading mw-heading2"><h2 id="Two-body_motion_is_planar">Two-body motion is planar</h2></div>

<p>The motion of two bodies with respect to each other always lies in a plane (in the <a href="Center_of_mass_frame" class="mw-redirect" title="Center of mass frame">center of mass frame</a>).

</p><p>Proof: Defining the <a href="Linear_momentum" class="mw-redirect" title="Linear momentum">linear momentum</a> <span class="texhtml"><b>p</b></span> and the <a href="Angular_momentum" title="Angular momentum">angular momentum</a> <span class="texhtml"><b>L</b></span> of the system, with respect to the center of mass, by the equations

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {L} =\mathbf {r} \times \mathbf {p} =\mathbf {r} \times \mu {\frac {d\mathbf {r} }{dt}},}</annotation>

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</p><p>where <span class="texhtml mvar" style="font-style:italic;">μ</span> is the <a href="Reduced_mass" title="Reduced mass">reduced mass</a> and <span class="texhtml"><b>r</b></span> is the relative position <span class="texhtml"><b>r</b><sub>2</sub> − <b>r</b><sub>1</sub></span> (with these written taking the center of mass as the origin, and thus both parallel to <span class="texhtml"><b>r</b></span>) the rate of change of the angular momentum <span class="texhtml"><b>L</b></span> equals the net <a href="Torque" title="Torque">torque</a> <span class="texhtml"><b>N</b></span>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} ={\frac {d\mathbf {L} }{dt}}={\dot {\mathbf {r} }}\times \mu {\dot {\mathbf {r} }}+\mathbf {r} \times \mu {\ddot {\mathbf {r} }}\ ,}</annotation>

</semantics>

</math></span></span>

and using the property of the <a href="Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a> that <span class="texhtml"><b>v</b> × <b>w</b> = <b>0</b></span> for any vectors <span class="texhtml"><b>v</b></span> and <span class="texhtml"><b>w</b></span> pointing in the same direction,

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {N} \ =\ {\frac {d\mathbf {L} }{dt}}=\mathbf {r} \times \mathbf {F} \ ,}">

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {N} \ =\ {\frac {d\mathbf {L} }{dt}}=\mathbf {r} \times \mathbf {F} \ ,}</annotation>

</semantics>

</math></span></span>

with <span class="texhtml"><b>F</b> = <i>μ</i> <i>d</i><span style="padding-left:0.12em;"><sup>2</sup></span><b>r</b>/<i>dt</i><span style="padding-left:0.12em;"><sup>2</sup></span></span>.

</p><p>Introducing the assumption (true of most physical forces, as they obey <a href="Newton's_laws_of_motion" title="Newton's laws of motion">Newton's strong third law of motion</a>) that the force between two particles acts along the line between their positions, it follows that <span class="texhtml"><b>r</b> × <b>F</b> = <b>0</b></span> and the <a href="Conservation_of_angular_momentum" class="mw-redirect" title="Conservation of angular momentum">angular momentum vector <span class="texhtml"><b>L</b></span> is constant</a> (conserved). Therefore, the displacement vector <span class="texhtml"><b>r</b></span> and its velocity <span class="texhtml"><b>v</b></span> are always in the plane <a href="Perpendicular" title="Perpendicular">perpendicular</a> to the constant vector <span class="texhtml"><b>L</b></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Energy_of_the_two-body_system">Energy of the two-body system</h2></div>

<p>If the force <span class="texhtml"><b>F</b>(<b>r</b>)</span> is <a href="Conservative_force" title="Conservative force">conservative</a> then the system has a <a href="Potential_energy" title="Potential energy">potential energy</a> <span class="texhtml"><i>U</i>(<b>r</b>)</span>, so the total <a href="Mechanical_energy" title="Mechanical energy">energy</a> can be written as

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

<mfrac>

</mfrac>

</mrow>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{\text{tot}}={\frac {1}{2}}m_{1}{\dot {\mathbf {x} }}_{1}^{2}+{\frac {1}{2}}m_{2}{\dot {\mathbf {x} }}_{2}^{2}+U(\mathbf {r} )={\frac {1}{2}}(m_{1}+m_{2}){\dot {\mathbf {R} }}^{2}+{1 \over 2}\mu {\dot {\mathbf {r} }}^{2}+U(\mathbf {r} )}</annotation>

</semantics>

</math></span></span>

</p><p>In the center of mass frame the <a href="Kinetic_energy#Frame_of_reference" title="Kinetic energy">kinetic energy</a> is the lowest and the total energy becomes

<mfrac>

</mfrac>

</mrow>

<msup>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E={\frac {1}{2}}\mu {\dot {\mathbf {r} }}^{2}+U(\mathbf {r} )}</annotation>

</semantics>

</math></span></span>

The coordinates <span class="texhtml"><b>x</b><sub>1</sub></span> and <span class="texhtml"><b>x</b><sub>2</sub></span> can be expressed as

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{1}={\frac {\mu }{m_{1}}}\mathbf {r} }</annotation>

</semantics>

</math></span></span>

<msub>

</mrow>

</mrow>

</msub>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{2}=-{\frac {\mu }{m_{2}}}\mathbf {r} }</annotation>

</semantics>

</math></span></span>

and in a similar way the energy <i>E</i> is related to the energies <span class="texhtml"><i>E</i><sub>1</sub></span> and <span class="texhtml"><i>E</i><sub>2</sub></span> that separately contain the kinetic energy of each body:

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

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</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mover>

</mrow>

</mover>

</mrow>

</mrow>

</mrow>

</msubsup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

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</mtd>

<mtd>

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</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{1}&={\frac {\mu }{m_{1}}}E={\frac {1}{2}}m_{1}{\dot {\mathbf {x} }}_{1}^{2}+{\frac {\mu }{m_{1}}}U(\mathbf {r} )\\[4pt]E_{2}&={\frac {\mu }{m_{2}}}E={\frac {1}{2}}m_{2}{\dot {\mathbf {x} }}_{2}^{2}+{\frac {\mu }{m_{2}}}U(\mathbf {r} )\\[4pt]E_{\text{tot}}&=E_{1}+E_{2}\end{aligned}}}</annotation>

</semantics>

</math></span></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="Central_forces">Central forces</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Classical_central-force_problem" title="Classical central-force problem">Classical central-force problem</a></div>

<p>For many physical problems, the force <span class="texhtml"><b>F</b>(<b>r</b>)</span> is a <a href="Central_force" title="Central force">central force</a>, i.e., it is of the form

</mrow>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\mathbf {r} )=F(r){\hat {\mathbf {r} }}}</annotation>

</semantics>

</math></span></span>

where <span class="texhtml"><i>r</i> = |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><b>r</b></span>|</span> and <span class="texhtml"><b>r̂</b> = <b>r</b>/<i>r</i></span> is the corresponding <a href="Unit_vector" title="Unit vector">unit vector</a>. We now have:

<mover>

</mrow>

</mover>

</mrow>

</mrow>

<mover>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu {\ddot {\mathbf {r} }}={F}(r){\hat {\mathbf {r} }}\ ,}</annotation>

</semantics>

</math></span></span>

where <span class="texhtml"><i>F</i>(<i>r</i>)</span> is negative in the case of an attractive force.

</p>

<ul><li><a href="Energy_drift" title="Energy drift">Energy drift</a></li>

<li><a href="Equation_of_the_center" title="Equation of the center">Equation of the center</a></li>

<li><a href="Euler's_three-body_problem" title="Euler's three-body problem">Euler's three-body problem</a></li>

<li><a href="Kepler_orbit" title="Kepler orbit">Kepler orbit</a></li>

<li><a href="Kepler_problem" title="Kepler problem">Kepler problem</a></li>

<li><a href="N-body_problem" title="N-body problem"><i>n</i>-body problem</a></li>

<li><a href="Three-body_problem" title="Three-body problem">Three-body problem</a></li>

<li><a href="Virial_theorem" title="Virial theorem">Virial theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFLuo2020" class="citation journal cs1">Luo, Siwei (22 June 2020). <a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F2399-6528%2Fab9c30">"The Sturm-Liouville problem of two-body system"</a>. <i>Journal of Physics Communications</i>. <b>4</b> (6): 061001. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020JPhCo...4f1001L">2020JPhCo...4f1001L</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F2399-6528%2Fab9c30">10.1088/2399-6528/ab9c30</a></span>.</cite></span>

</li>

<li id="cite_note-Betounes-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Betounes_2-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFDavid_Betounes2001" class="citation book cs1">David Betounes (2001). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/differentialequa0000beto"><i>Differential Equations</i></a></span>. Springer. p. 58; Figure 2.15. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-95140-7</bdi>.</cite></span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFLandau_LDLifshitz_EM1976" class="citation book cs1"><a href="Lev_Landau" title="Lev Landau">Landau LD</a>; <a href="Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz EM</a> (1976). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mechanics00land"><i>Mechanics</i></a></span> (3rd. ed.). New York: <a href="Pergamon_Press" title="Pergamon Press">Pergamon Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-08-029141-4</bdi>.</cite></li>

<li><cite id="CITEREFGoldstein_H1980" class="citation book cs1"><a href="Herbert_Goldstein" title="Herbert Goldstein">Goldstein H</a> (1980). <i><a href="Classical_Mechanics_(Goldstein)" title="Classical Mechanics (Goldstein)">Classical Mechanics</a></i> (2nd. ed.). New York: <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-201-02918-9</bdi>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://scienceworld.wolfram.com/physics/Two-BodyProblem.html">Two-body problem</a> at <a href="ScienceWorld" class="mw-redirect" title="ScienceWorld">Eric Weisstein's World of Physics</a></li></ul>

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