# Introduction

The three body problem is one of the oldest problems in dynamical systems. The problem was designed to determine the behavior of three interacting masses. In the circular restricted three body problem we consider a case where one of the bodies is so much less massive than that of the other two that the behavior of the other more massive bodies are unaffected by its gravitational potential. In addition we constrain the two primary (massive) bodies to rotate about their common center of mass in circles. This scenario is good for modeling the trajectories of spacecraft in the gravitational potential of two massive bodies. http://www.geom.uiuc.edu/~megraw/CR3BP_html/cr3bp_bg.html

Joseph-Louis Lagrange showed that there were at least some solutions to the three body problem if we restricted the three bodies to move in the same plane and assumed that the mass of one of them was so small as to be negligible. In his solutions, the three bodies move in unison, always maintaining the same positions relative to each other. http://plus.maths.org/issue6/xfile/index.html

NB Communication has no/negliable MASS

# n-body problem

The correct name for the three body problem

The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e., Newton's laws of motion and Newton's law of gravity. http://en.wikipedia.org/wiki/N-body_problem

# 3 Manifold Theory

Phenomena in three dimensions can be strikingly different from that for other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions greater than three. Perhaps surprisingly, this special role has led to the discovery of close connections to a diversity of other fields, such as knot theory, geometric group theory, hyperbolic geometry, number theory, Teichmüller theory, topological quantum field theory, gauge theory, Floer homology, and partial differential equations. 3-manifold theory is considered a part of low-dimensional topology or geometric topology. http://en.wikipedia.org/wiki/3-manifold

# References

**Full details**

http://www.geom.uiuc.edu/~megraw/CR3BP_html/cr3bp_bg.html