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Title:  Instability of Periodic Orbits of Some Rhombus and Parallelogram Four Body Problems 
Authors:  Mansur, ABDALLA 

Keywords:  NBody Problem 
Issue Date:  27Nov2012 
Series/Report no.:  Canadian theses 
Abstract:  The rhombus and parallelogram orbits are interesting families of periodic solutions, which come from celestial mechanics and the Nbody problem. Variational methods with finite order symmetry group are used to construct minimizing noncollision periodic orbits.
We study the question of stability or instability of periodic and symmetric periodic solutions of the rhombus and the equal mass parallelogram four body problems. We first study the stability of periodic solutions for the rhombus four body problem. An analytical description of the variational principle is used to show that the homographic solutions are the minimizers of the action functional restricted to rhombus loop space [23]. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing rhombus orbit to prove the main result, Theorem 4.2.2, which states that the reduced rhombus orbit is hyperbolic in the reduced energy manifold when it is not degenerate.
We second study the stability for symmetric periodic solutions of the equal mass parallelogram four body problem. The parallelogram family is a family of Z_2× Z_4 symmetric action minimizing solutions, investigated by [7]. In this example, the minimizing solution [7] can be extended to a 4Tperiodic solution using symmetries through square and collinear configurations. The Maslov index of the orbits is used to prove the main result, Theorem 5.3.1, which states that the minimizing equal mass parallelogram solution is unstable when it is nondegenerate. 
Description:  Thesis (Ph.D, Mathematics & Statistics)  Queen's University, 20121126 11:30:29.688 
URI:  http://hdl.handle.net/1974/7650 
Appears in Collections:  Queen's Theses & Dissertations Mathematics & Statistics Graduate Theses

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