Variational approach to second species periodicsolutions of Poincaré of the 3 body problem

Sergey V. Bolotin; Piero Negrini

doi:10.3934/dcds.2013.33.1009

Article Contents

2013, Volume 33, Issue 3: 1009-1032. Doi: 10.3934/dcds.2013.33.1009

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Variational approach to second species periodic solutions of Poincaré of the 3 body problem

Sergey V. Bolotin^1, and
Piero Negrini^2,

1.
Department of Mathematics, University of Wisconsin, Madison
2.
Department of Mathematics, La Sapienza, University of Rome

Received: March 31, 2011

Revised: January 31, 2012

Published: February 2013

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Abstract

We consider the plane 3 body problem with 2 of the masses small. Periodic solutions with near collisions of small bodies were named by Poincaré second species periodic solutions. Such solutions shadow chains of collision orbits of 2 uncoupled Kepler problems. Poincaré only sketched the proof of the existence of second species solutions. Rigorous proofs appeared much later and only for the restricted 3 body problem. We develop a variational approach to the existence of second species periodic solutions for the nonrestricted 3 body problem. As an application, we give a rigorous proof of the existence of a class of second species solutions.

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Mathematics Subject Classification: 70F10, 70F15, 37N05.

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