# American Institute of Mathematical Sciences

March  2013, 33(3): 1137-1155. doi: 10.3934/dcds.2013.33.1137

## 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, Canada 2 Faculty of Science, University of Ontario Institute of Technology, Oshawa, Ontario L1H 7K4, Canada, Canada

Received  April 2011 Revised  March 2012 Published  October 2012

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.
Citation: Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete & Continuous Dynamical Systems, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137
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##### References:
 [1] Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002 [2] Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001 [3] Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463 [4] Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214 [5] Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014 [6] Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044 [7] Rongchang Liu, Jiangyuan Li, Duokui Yan. New periodic orbits in the planar equal-mass three-body problem. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2187-2206. doi: 10.3934/dcds.2018090 [8] Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015 [9] Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 [10] Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199 [11] Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069 [12] Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091 [13] Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034 [14] Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213 [15] Xiaoni Chi, Zhongping Wan, Zijun Hao. A full-modified-Newton step $O(n)$ infeasible interior-point method for the special weighted linear complementarity problem. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021082 [16] Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic & Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006 [17] Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $C^{1}$ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002 [18] Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008 [19] Jinye Shen, Xian-Ming Gu. Two finite difference methods based on an H2N2 interpolation for two-dimensional time fractional mixed diffusion and diffusion-wave equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021086 [20] Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021013

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