n-body Problem

n-body Problem

The -body problem is the problem of predicting the motion of a group of celestial objects that interact with each other gravitationally. Solving this problem has been motivated by the need to understand the motion of the Sun, planets and the visible stars. Its first complete mathematical formulation appeared in Isaac Newton's Principia (the -body problem in general relativity is considerably more difficult). Since gravity was responsible for the motion of planets and stars, Newton had to express gravitational interactions in terms of differential equations. Newton proved in the Principia that a spherically-symmetric body can be modelled as a point mass.

The 2-body problem has been completely solved. For n = 3, solutions exist for special cases. A general solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary n can be given in terms of Taylor series, but in practice such an infinite series must be truncated, giving an approximate solution. In addition, many solutions by numerical integration exist, but these too are approximate solutions.