Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses

Mark Lewis; Daniel Offin; Pietro-Luciano Buono; Mitchell Kovacic

doi:10.3934/dcds.2013.33.1137

Article Contents

2013, Volume 33, Issue 3: 1137-1155. Doi: 10.3934/dcds.2013.33.1137

This issue Previous Article Discrete dynamics in implicit form Next Article On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$

Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses

1.
Department of Mathematics, Queen's University, Kingston, Ontario K7L 4V1
2.
Faculty of Science, University of Ontario Institute of Technology, Oshawa, Ontario L1H 7K4

Received: March 31, 2011

Revised: February 29, 2012

Published: February 2013

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Abstract

The hip-hop orbit is an interesting symmetric periodic family of orbits whereby the global existence methods of variational analysis applied to the N-body problem result in a collision free solution of (1). Perturbation techniques have been applied to study families of hip-hop like orbits bifurcating from a uniformly rotating planar 2N-gon [4] with equal masses, or a uniformly rotating planar 2N+1 body relative equilibrium with a large central mass [18]. We study the question of stability or instability for symmetric periodic solutions of the equal mass $2N$-body problem without perturbation methods. The hip-hop family is a family of $\mathbb{Z}_2$-symmetric action minimizing solutions, investigated by [7,23], and is shown to be generically hyperbolic on its reduced energy-momentum surface. We employ techniques from symplectic geometry and specifically a variant of the Maslov index for curves of Lagrangian subspaces along the minimizing hip-hop orbit to develop conditions which preclude the existence of eigenvalues of the monodromy matrix on the unit circle.

Keywords:

Hamiltonian,
n-body problem,
equivariant action integral,
reduced space,
spatial and time reversing symmetry group,
Lagrangian plane,
Maslov index.

Mathematics Subject Classification: 34C35, 34C27, 54H20.

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