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Families of double symmetric “Schubart-like” periodic orbits

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Abstract

We consider a class of Hamiltonian systems with two degrees of freedom with singularities. This class includes several symmetric subproblems of the \(n\)-body problem where the singularities are due to collisions involving two or more bodies. “Schubart-like” periodic orbits having two collisions in one period, are present in most of these subproblems. The purpose of this paper is to study the existence of families of such a periodic orbits in a general setting. The blow up techniques of total collision and infinity are applied to our class of Hamiltonian system. This allows us to derive sufficient conditions to ensure the existence of families of double symmetric “Schubart-like” periodic orbits having many singularities. The orbits in the family can be parametrized by the number of singularities in one period. The results are applied to some subproblems of the gravitational \(n\)-body problem.

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Acknowledgments

This work has been supported by grants MTM2006-05849/Consolider, MTM2010-16425 (Spain) and CIRIT 2008 SGR-67 (Catalonia). The author is grateful to Carles Simó for useful comments.

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  1. Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    Regina Martínez

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  1. Regina Martínez
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Correspondence to Regina Martínez.

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Martínez, R. Families of double symmetric “Schubart-like” periodic orbits. Celest Mech Dyn Astr 117, 217–243 (2013). https://doi.org/10.1007/s10569-013-9509-4

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