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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.33 The Three-Body Problem

The three-body problem can be simply stated: three point masses move in space under their mutual gravitational attraction; given their initial positions and velocities, determine their subsequent motion. Initially, it may come as a surprise that this is a difficult problem, since the analogous two-body problem can be solved fairly simply: more precisely, given any set of initial conditions, we can write down a formula, in terms of elementary functions (these are functions that can be built up using the basic operations of arithmetic, together with a few standard functions such as the EXPONENTIAL [III.25] and TRIGONOMETRIC [III.92] functions), that tells us the subsequent positions and velocities of the bodies. However, ...

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