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 .

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{q}_j(0)

and velocities

\dot\mathbf{q}_j(0)

of ''n'' particles (''j''=1,...,''n'') with no

\mathbf{q}_j(0) 
eq \mathbf{q}_k(0)

for mutually distinct ''j '' and ''k '', find the solution of the second order system

m_j	\ddot{\mathbf{q}}_j	= \gamma \sum\limits_{k
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 cannot be regularised analytically, hence Sundman's regularization cannot 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

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Singularities of the ''n''-body problem

For details see No-collisions Singularities . Basically there can be two types of singularities of the n-body problem:

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

singularities in which a collapse does not occur, but q(t) does not remain finite. The latter one are called No-collisions Singularities . Their existence has been conjectured for ''n>3'' by Painlevé (see Painlevé's Conjecture ).

TRIVIA

The three-body problem is figured prominently in the '' Criminal Minds '' television series episode "Compulsion."

The n-body problem also appears on the 1951 Science Fiction movie '' The Day The Earth Stood Still '', where Klaatu solves it in order to attract a scientist's attention.

	CATEGORIES ABOUT N-BODY PROBLEM

	celestial mechanics

	classical mechanics

	problems

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Three-body Problem