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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 845795, 5 pages
http://dx.doi.org/10.1155/2013/845795
Research Article

The Characterization of the Variational Minimizers for Spatial Restricted -Body Problems

Yangtze Center of Mathematics and College of Mathematics, Sichuan University, Chengdu 610064, China

Received 8 February 2013; Accepted 27 April 2013

Academic Editor: Maoan Han

Copyright © 2013 Fengying Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We use Jacobi's necessary condition for the variational minimizer to study the periodic solution for spatial restricted -body problems with a zero mass on the vertical axis of the plane for equal masses. We prove that the minimizer of the Lagrangian action on the anti-T/2 or odd symmetric loop space must be a nonconstant periodic solution for any ; hence the zero mass must oscillate, so that it cannot be always in the same plane with the other bodies. This result contradicts with our intuition that the small mass should always be at the origin.