American Institute of Mathematical Sciences

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September  2008, 10(2&3, September): 699-717. doi: 10.3934/dcdsb.2008.10.699

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, France

Received  October 2006 Revised  August 2007 Published  June 2008

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.
Keywords: Periodic Solutions., Three-Body Problem, Celestial Mechanics.
Mathematics Subject Classification: Primary: 70F07, 70F15; Secondary: 37J5.
Citation: Andrea Venturelli. A Variational proof of the existence of Von Schubart's orbit. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 699-717. doi: 10.3934/dcdsb.2008.10.699
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