Mechanics of particles and systems Particle systems and celestial mechanics Problem

Three-body problem

Builds on Ordinary differential equation Hamiltonian system n-body problem

Problem

Let $G > 0$ be the gravitational constant and let $m_1, m_2, m_3 > 0$ be masses. Let the positions $x_1(t), x_2(t), x_3(t) \in \R^3$ satisfy Newton's equations $x_i'' = \sum_{j \ne i} \frac{G m_j (x_j - x_i)}{\norm{x_j - x_i}^3}, \qquad i = 1, 2, 3.$ Given distinct initial positions and initial velocities, describe the solution for all time.

Remarks

This is the case $n = 3$ of the n-body problem. Bruns showed in 1887 that the ten classical first integrals are the only independent algebraic ones. Poincaré 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. Sundman 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.

Special periodic solutions exist. Euler 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. For three equal masses there is a solution in which the bodies chase each other along a figure-eight curve. Moore found it numerically in 1993, and Chenciner and Montgomery proved its existence in 2000.

In 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. In a rotating frame it is a Hamiltonian system with two degrees of freedom.

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Tag	`JE1Z920QKT`
URI	`m2://JE1Z920QKT`
Kind	problem
Writing	W0 stub
MSC	70F07, 37N05, 70F15
Status	This copy of the site has no build data. Run `m2 status JE1Z920QKT`.
About	n-body problem
Uses	Ordinary differential equation, Hamiltonian system, n-body problem
Same as	Wikidata Q607699 Wikipedia Three-body problem

Three-body problem

Problem

Remarks

Reading path

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