---
tag: 62SNWC5E45
kind: problem
name: n-body problem
msc: [70F10, 37N05, 70F15]
uses: [0NT64JV96A, 0EDKGA2FAK, 0262JP0NV9]
about: [0EDKGA2FAK]
sources:
  - url: https://en.wikipedia.org/wiki/N-body_problem
same_as:
  wikidata: Q1199050
  wikipedia: N-body problem
history:
  posed_by: "Isaac Newton"
  first_stated: "1687"
---

## Problem

Let $n \ge 2$, let $G > 0$ be the gravitational constant, and let $m_1, \ldots, m_n > 0$ be masses.
Let the positions $x_1(t), \ldots, x_n(t) \in \R^3$ satisfy Newton's equations
$$m_i \, x_i'' = \sum_{j \ne i} \frac{G m_i m_j (x_j - x_i)}{\norm{x_j - x_i}^3}, \qquad i = 1, \ldots, n.$$
Given the initial positions, with $x_i(0) \ne x_j(0)$ for $i \ne j$, and the initial velocities, describe the solution for all time.

## Remarks

The equations are a system of [[ordinary differential equations|0NT64JV96A]] of order two.
They form a [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian
$$H = \sum_i \frac{\norm{p_i}^2}{2 m_i} - \sum_{i < j} \frac{G m_i m_j}{\norm{x_i - x_j}}, \qquad p_i = m_i x_i'.$$
There are ten classical [[first integrals|0262JP0NV9]].
They are the energy, the three components of the total momentum, the three components of the angular momentum, and three from the uniform motion of the centre of mass.

For $n = 2$ the problem reduces to the Kepler problem and is solved completely: the relative motion follows a conic section.
For $n \ge 3$ there is no general solution by elementary formulas, and the [[three-body problem|JE1Z920QKT]] already shows chaotic behaviour.
A solution can stop existing in finite time at a singularity.
Collisions are one kind of singularity.
Xia proved in 1992 that for $n = 5$ there are also singularities without collision, in which bodies escape to infinity in finite time.

## Explanations

### L1 human

Put several stars or planets in space, give them initial positions and speeds, and let gravity act.
The $n$-body problem asks how they move from then on.
With two bodies the answer is the ellipses, parabolas and hyperbolas of Kepler.
With three or more bodies the motion can be extremely complicated, and much of the theory of dynamical systems grew out of attempts to understand it.