# American Institute of Mathematical Sciences

September  2008, 10(2&3, September): 609-620. doi: 10.3934/dcdsb.2008.10.609

## A topological existence proof for the Schubart orbits in the collinear three-body problem

 1 School of Mathematics, University of Minnesota, Minneapolis MN 55455, United States

Received  November 2006 Revised  June 2007 Published  June 2008

A topological existence proof is presented for certain symmetrical periodic orbits of the collinear three-body problem with two equal masses, called Schubart orbits. The proof is based on the construction of a Wazewski set $W$ in the phase space. The periodic orbits are found by a shooting argument in which symmetrical initial conditions entering $W$ are followed under the flow until they exit $W$. Topological considerations show that the image of the symmetrical entrance states under this flow map must intersect an appropriate set of symmetrical exit states.
Citation: Richard Moeckel. A topological existence proof for the Schubart orbits in the collinear three-body problem. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 609-620. doi: 10.3934/dcdsb.2008.10.609
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