Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof

Pages: 1763 - 1774, Volume 32, Issue 5, May 2012

doi:10.3934/dcds.2012.32.1763       Abstract        References        Full Text (383.1K)       Related Articles

Hsin-Yuan Huang - School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States (email)

Abstract: The Schubart-like orbits in the collinear four-body problem are similar to those discovered numerically by Schubart[12] in the collinear three-body problem. Schubart-like orbits are periodic solutions with exactly two binary collisions and one simultaneous binary collision per period. The proof of the existence of these orbits given in this paper is based on the direct method of Calculus of Variations. We exploit the variational structure of the problem and show that the minimizers of the Lagrangian action functional in a suitably chosen space have the desired properties.

Keywords:  Four-body problem, periodic solutions, variational methods.
Mathematics Subject Classification:  Primary: 70F16, 70F10; Secondary: 35A15.

Received: December 2010;      Revised: May 2011;      Published: January 2012.

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