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The two-body problem considers two rigid point masses in mutual orbit about each other. To determine the motion of
these bodies, first find the vector equations of motion. Given two bodies with masses  and  , let
 be the
vector from the center of mass to and
 be the vector from the center of mass to . From the definition
of center of mass
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(1) |
Define
where  is the so-called reduced mass. The vector displacement from to is
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(4) |
The distance between the two mutually orbiting bodies is
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(5) |
Equations (1)-(5) lead to the identities
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(6) |
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(7) |
so
In terms of r,
The differential equations of motion are then
Plugging (10)-(11) into (12)-(13),
or
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(16) |
 Circular Orbit, Elliptical Orbit, Hyperbolic Orbit, Kepler Problem, Many-Body Problem, n-Body Problem, Orbit, Parabolic Orbit, Relativistic
Two-Body Problem, Restricted Three-Body Problem, Three-Body Problem
© 1996-2007 Eric W. Weisstein
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