The two-body problem can be re-formulated as two **one-body problems**, a trivial one and one that involves solving for the motion of one particle in an external potential. Since many one-body problems can be solved exactly, the corresponding two-body problem can also be solved. By contrast, the three-body problem (and, more generally, the *n*-body problem for *n* ≥ 3) cannot be solved in terms of first integrals, except in special cases.

Let **x**_{1} and **x**_{2} be the vector positions of the two bodies, and *m*_{1} and *m*_{2} be their masses. The goal is to determine the trajectories **x**_{1}(*t*) and **x**_{2}(*t*) for all times *t*, given the initial positions **x**_{1}(*t* = 0) and **x**_{2}(*t* = 0) and the initial velocities **v**_{1}(*t* = 0) and **v**_{2}(*t* = 0).

When applied to the two masses, Newton's second law states that

where **F**_{12} is the force on mass 1 due to its interactions with mass 2, and **F**_{21} is the force on mass 2 due to its interactions with mass 1. The two dots on top of the **x** position vectors denote their second derivative with respect to time, or their acceleration vectors.

Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. *Adding* equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, *subtracting* equation (2) from equation (1) results in an equation that describes how the vector **r** = **x**_{1} − **x**_{2} between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories **x**_{1}(*t*) and **x**_{2}(*t*).

This page was last edited on 29 April 2018, at 13:49 (UTC).

Reference: https://en.wikipedia.org/wiki/Two-body_problem under CC BY-SA license.

Reference: https://en.wikipedia.org/wiki/Two-body_problem under CC BY-SA license.

- Classical Mechanics
- Satellite
- Planet
- Planet
- Star
- Two
- Stars
- Binary Star
- Electron
- Atomic Nucleus
- Potential
- Three-body Problem
- N-body Problem
- Newton's Second Law
- Center Of Mass
- Barycenter

- Two-body Problem
- Caroline Van Hook Bean
- Ganda, Angola
- Bolivia National Football Team
- National Park Service
- Arthur Milton
- Black Forest National Park
- Floppy Disk Drive
- Peter Burge (cricketer)
- Johnny Douglas
- Darius I
- Appalachian Trail
- New York - New Jersey Trail Conference
- Mount Washington State Forest
- Alain Badiou
- McEvedy Shield
- United States Geological Survey
- Clean Sheet
- Qeshm, Iran
- Jimmy Douglas (American Soccer)
- Lucien Laurent
- Polycistronic
- LIG4
- Stamp Seal
- Hello Tomorrow
- Bash Bish Falls State Park
- Mount Everett State Reservation
- Copake Falls, New York
- Permissive Dialling
- Cedar Grove Cemetery (Queens, New York)