Resumen de Instability for a Family of Homographic Periodic Solutions in the Parallelogram Four Body Problem

Abdalla Mansur, Daniel Offin, Mark Lewis

• We consider the question of stability for the homographic family of rhombus solutions within the four degree of freedom parallelogram four body problem. Our approach to demonstrate spectral instability for the entire parameter ranges of mass ratio and eccentricity e ≥ 0 , rests on a topological invariant which is the Maslov index. The analysis begins with a holonomically constrained system of rhombus configurations.

  This is a three degree of freedom problem and the homographic solutions we are studying belong to the constraint set as well as the unconstrained parallelogram system. Minimizing the constrained action functional over a homology class of rhombus loops brings us to the homographic family. After reducing by the angular momentum to remove degenerate eigenvalues of the Poincaré map, we use the minimization property together with an analysis of Lagrangian subspaces in the unreduced space to demonstrate the hyperbolicity (assuming nondegeneracy of the variational problem) on the reduced energy surface of the constrained system. In the last section we argue that the unconstrained homographic solutions of the parallelogram four body problem inherit this nonlinear instability.