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A variational proof of existence of transit orbits in the restricted three-body problem. (English) Zbl 1061.70005

Summary: Because of the Jacobi integral, solutions of the planar, circular restricted three-body problem are confined to certain subsets of the plane called Hill’s regions. For certain values of the integral, one component of the Hill’s region consists of disk-like regions around of the two primary masses, connected by a tunnel near the collinear Lagrange point, \(L_2\). A ‘transit orbit’ is a solution which crosses the tunnel, in a sense which can be made precise using Conley’s isolating block construction [C. C. Conley, SIAM J. Appl. Math. 16, 732–746 (1968; Zbl 0197.21105)]. For values of the Jacobi integral sufficiently close to its value at \(L_2\), Conley found transit orbits by linearizing near the equilibrium point. The goal of this paper is to develop a method for proving existence of transit orbits for values of Jacobi constant far from equilibrium. The method is based on Maupertuis variational principle, but isolating blocks turn out to play an important role.

MSC:

70F07 Three-body problems
70H30 Other variational principles in mechanics
70G75 Variational methods for problems in mechanics

Citations:

Zbl 0197.21105
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References:

[1] DOI: 10.1137/0116060 · Zbl 0197.21105 · doi:10.1137/0116060
[2] DOI: 10.1016/0022-0396(70)90123-3 · Zbl 0185.27501 · doi:10.1016/0022-0396(70)90123-3
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