N-body problem Totally Explained

Everything about Three Body Problem totally explained

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, for example, Newton's laws of motion and Newton's law of gravity.

Mathematical formulation of the n-body problem

The general n-body problem of celestial mechanics is an initial-value problem for ordinary differential equations. Given initial values for the positions

mathbf.

This finishes the proof of Sundman's theorem. Unfortunately the corresponding convergent series converges very slowly. That is, getting the value to any useful precision requires so many terms that his solution is of little practical use.

The global solution of the n-body problem

In order to generalise Sundman's result for the case n>3 (or n=3 and c=0) one has to face two obstacles:

As it has been shown by Siegel, that collisions which involve more than 2 bodies can't be regularised analytically, hence Sundman's regularization can't be generalised.

The structure of singularities is more complicated in this case, other types of singularities may occur. Finally Sundman's result was generalised to the case of n>3 bodies by Q. Wang in the 1990s. Since the structure of singularities is more complicated, Wang had to leave out completely the questions of singularities. The central point of his approach is to transform, in an appropriate manner, the equations to a new system, such that the interval of existence for the solutions of this new system is

[0,infty)

Singularities of the n-body problem

n-body problem:

collisions of one, two or n particles, but for which q(t) remains finite.

singularities in which a collapse doesn't occur, but q(t) doesn't remain finite. The latter one are called no-collisions singularities. Their existence has been conjectured for n > 3 by Painlevé (see Painlevé's conjecture). Examples of this behaviour have been constructed by Xia and Gerver.Further Information

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