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

Hsin-Yuan Huang

doi:10.3934/dcds.2012.32.1763

Article Contents

2012, Volume 32, Issue 5: 1763-1774. Doi: 10.3934/dcds.2012.32.1763

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

Hsin-Yuan Huang^1,

1.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455

Received: November 30, 2010

Revised: April 30, 2011

Published: April 2012

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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.

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Mathematics Subject Classification: Primary: 70F16, 70F10; Secondary: 35A15.

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