Two-Body Problem -- from Eric Weisstein's World of Physics

Two-Body Problem

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

(1)

Define

(2)

(3)

where is the so-called reduced mass. The vector displacement from to is

(4)

The distance between the two mutually orbiting bodies is

(5)

Equations (1)-(5) lead to the identities

(6)

(7)

so

(8)

(9)

In terms of r,

(10)

(11)

The differential equations of motion are then

(12)

(13)

Plugging (10)-(11) into (12)-(13),

(14)

(15)

or

(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

References

Basdevant, J.-L. and Dalibard, J. "The Two-Body Problem." §9.1 in The Quantum Mechanics Solver: How to Apply Quantum Theory to Modern Physics. Berlin: Springer-Verlag, p. 61, 2000.