# N-body problem

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The n-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 n-body problem in general relativity is considerably more difficult).[citation needed] 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.

## Contents

### Informal version of the Newton n-body problem

The physical problem can be informally stated as:

More precisely,

In mathematical terms, this means to find a global solution of the initial value problem for the differential equations describing the n-body problem.

### 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 $\mathbf{q}_j(0) \neq \mathbf{q}_k(0)$ for all mutually distinct j and k , find the solution of the second order system