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  "body": "\n## Problem\n\nLet $n \\ge 2$, let $G \u003e 0$ be the gravitational constant, and let $m_1, \\ldots, m_n \u003e 0$ be masses.\nLet the positions $x_1(t), \\ldots, x_n(t) \\in \\R^3$ satisfy Newton's equations\n$$m_i \\, x_i'' = \\sum_{j \\ne i} \\frac{G m_i m_j (x_j - x_i)}{\\norm{x_j - x_i}^3}, \\qquad i = 1, \\ldots, n.$$\nGiven 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.\n\n## Remarks\n\nThe equations are a system of [[ordinary differential equations|0NT64JV96A]] of order two.\nThey form a [[Hamiltonian system|0EDKGA2FAK]] with Hamiltonian\n$$H = \\sum_i \\frac{\\norm{p_i}^2}{2 m_i} - \\sum_{i \u003c j} \\frac{G m_i m_j}{\\norm{x_i - x_j}}, \\qquad p_i = m_i x_i'.$$\nThere are ten classical [[first integrals|0262JP0NV9]].\nThey 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.\n\nFor $n = 2$ the problem reduces to the Kepler problem and is solved completely: the relative motion follows a conic section.\nFor $n \\ge 3$ there is no general solution by elementary formulas, and the [[three-body problem|JE1Z920QKT]] already shows chaotic behaviour.\nA solution can stop existing in finite time at a singularity.\nCollisions are one kind of singularity.\nXia proved in 1992 that for $n = 5$ there are also singularities without collision, in which bodies escape to infinity in finite time.\n\n## Explanations\n\n### L1 human\n\nPut several stars or planets in space, give them initial positions and speeds, and let gravity act.\nThe $n$-body problem asks how they move from then on.\nWith two bodies the answer is the ellipses, parabolas and hyperbolas of Kepler.\nWith 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.\n",
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