A Variational proof of the existence of Von Schubart's orbit

Andrea Venturelli

doi:10.3934/dcdsb.2008.10.699

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2008, Volume 10, Issue 2&3: 699-717. Doi: 10.3934/dcdsb.2008.10.699

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A Variational proof of the existence of Von Schubart's orbit

Andrea Venturelli^1,

1.
Laboratoire d'Analyse non-linéaire et géométrie, Université d'Avignon et des pays de Vaucluse, 33, Rue Louis Pasteur, 84000 Avignon

Received: October 2006

Revised: August 2007

Published: September 2008

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Abstract

Weconsiderthecollinearthree-bodyproblemwithtwoequalmasses for the Newtonian potential $1/r$. We give a rigorous proof of the existence of a symmetric periodic solution with two collisions per period. This solution has been discovered numerically in 1956 by J. von Schubart (see [12]). Our proof is based on the direct method in Calculus of Variations, which consists in the minimization of the action on a well chosen set of periodic loops. The main difficulty is to show that the minimizer has only two collisions per period.

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Mathematics Subject Classification: Primary: 70F07, 70F15; Secondary: 37J50.

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