Stability of the Laplace solutions of the unrestricted three-body problem

Astronomy and Astrophysics – Astronomy

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Celestial Mechanics, Laplace Equation, Motion Stability, Numerical Stability, Three Body Problem, Liapunov Functions

Scientific paper

A proof of the stability of the Laplace solutions of the plane unrestricted three-body problem in the case where interaction forces between two of the bodies are proportional to the distance between them, raised to a power greater than one, is presented. Consideration is given to the constant Laplacians for the motion of three bodies situated at the vertices of an equilateral triangle rotating uniformly about its center of mass, expressed in Liapunov variables. Making use of the theorem of Routh (1977) concerning the Liapunov stability of the constant Laplacians of the unrestricted three-body problem, it is shown that perturbed motion is limited to some given domain. It is also noted that when the interaction force is proportional to the first power of the distance between bodies, the equations of perturbed motion for the triangular libration points of the restricted three-body problem are integrable.

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