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  "body": "\n## Problem\n\nLet $G \u003e 0$ be the gravitational constant and let $m_1, m_2, m_3 \u003e 0$ be masses.\nLet the positions $x_1(t), x_2(t), x_3(t) \\in \\R^3$ satisfy Newton's equations\n$$x_i'' = \\sum_{j \\ne i} \\frac{G m_j (x_j - x_i)}{\\norm{x_j - x_i}^3}, \\qquad i = 1, 2, 3.$$\nGiven distinct initial positions and initial velocities, describe the solution for all time.\n\n## Remarks\n\nThis is the case $n = 3$ of the [[n-body problem|62SNWC5E45]].\nBruns showed in 1887 that the ten classical first integrals are the only independent algebraic ones.\nPoincaré found that the problem is not integrable and that its orbits can behave in a very complicated way, near homoclinic orbits; this was the beginning of chaos theory.\nSundman found in 1912 a convergent power series solution for all initial conditions with nonzero angular momentum, but it converges too slowly to be of practical use.\n\nSpecial periodic solutions exist.\nEuler found solutions with the three bodies on a rotating line, and Lagrange found solutions with the three bodies at the vertices of a rotating equilateral triangle.\nFor three equal masses there is a solution in which the bodies chase each other along a figure-eight curve.\nMoore found it numerically in 1993, and Chenciner and Montgomery proved its existence in 2000.\n\nIn the planar circular restricted three-body problem one body has negligible mass and moves in the plane of the other two, which move on circles.\nIn a rotating frame it is a [[Hamiltonian system|0EDKGA2FAK]] with two degrees of freedom.\n\n## Explanations\n\n### L1 human\n\nThree bodies, such as the Sun, the Earth and the Moon, attract each other by gravity.\nThe problem asks how they move.\nUnlike the case of two bodies, there is no general formula: tiny changes of the starting positions can lead to completely different motions.\nYet there are beautiful special motions, like three bodies at the corners of a rotating triangle, and the search for them shaped the study of [[ordinary differential equations|0NT64JV96A]].\n",
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