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 - A, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137
References:
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References:
 [1] V. I. Arnol'd, Characteristic class entering in quantization conditions,, Funct. Anal. Appl., 1 (1967), 1.  doi: 10.1007/BF01075861.  Google Scholar [2] V. I. Arnol'd, "Dynamical Systems III,", Encyclopaedia Math. Sci., 3 (1988).   Google Scholar [3] V. I. Arnol'd, Sturm theorems and symplectic geometry,, Funct. Anal. Appl., 19 (1985), 251.   Google Scholar [4] E. Barrabés, J. M. Cors, C. Pinyol and J. Soler, Hip-hop solutions of the 2N-body problem,, Celest. Mech. Dynam. Astron., 95 (2006), 55.  doi: 10.1007/s10569-006-9016-y.  Google Scholar [5] P. L. Buono, M. Kovacic, M. Lewis and D. Offin, Symmetry-breaking bifurcations of the hip-hop orbit,, in preparation., ().   Google Scholar [6] A. Chenciner and R. Montgomery, On a remarkable periodic orbit of the three body problem in the case of equal masses,, Ann. Math., 152 (2000), 881.  doi: 10.2307/2661357.  Google Scholar [7] A. Chenciner and A. Venturelli, Minima de l'intégrale d'action du problème Newtonien de 4 corps de masses égales dans $\mathbbR^3$: orbites 'hip-hop',, Celest. Mech. Dynam. Astron., 77 (2000), 139.  doi: 10.1023/A:1008381001328.  Google Scholar [8] G. F. Dell'Antonio, Variational calculus and stability of periodic solutions of a class of Hamiltonian systems,, Reviews in Math. Physics, 6 (1994), 1187.  doi: 10.1142/S0129055X94000432.  Google Scholar [9] J. J. Duistermaat, On the morse index in variational calculus,, Adv. Math., 21 (1976), 173.  doi: 10.1016/0001-8708(76)90074-8.  Google Scholar [10] I. Ekeland, "Convexity Methods in Hamiltonian Systems,", Springer-Verlag, (1991).   Google Scholar [11] D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem,, Inv. Math., 155 (2004), 305.  doi: 10.1007/s00222-003-0322-7.  Google Scholar [12] W. B. Gordon, A minimizing property of Keplerian orbits,, Amer. J. Math., 99 (1970), 961.  doi: 10.2307/2373993.  Google Scholar [13] X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian systems with application to figure-eight orbit,, Comm. Math. Phys., 290 (2009), 737.  doi: 10.1007/s00220-009-0860-y.  Google Scholar [14] J. Marsden, "Lectures on Mechanics,", Springer-Verlag, (1991).   Google Scholar [15] C. Marchal, How the method of minimization of action avoids singularities,, Celest. Mech. Dynam. Astron., 83 (2002), 325.  doi: 10.1023/A:1020128408706.  Google Scholar [16] V. P. Maslov, "Theory of Perturbations and Asymptotic Methods,", (Russian), (1965).   Google Scholar [17] K. R. Meyer, Hamiltonian systems with a discrete symmetry,, J. Diff. Eqns., 41 (1981), 228.  doi: 10.1016/0022-0396(81)90059-0.  Google Scholar [18] K. R. Meyer and D. S. Schmidt, Librations of central configurations and braided Saturn rings,, Celest. Mech. Dynam. Astron., 55 (1993), 289.  doi: 10.1007/BF00692516.  Google Scholar [19] D. C. Offin, Hyperbolic minimizing geodesics,, Trans. Amer. Math. Soc., 352 (2000), 3323.  doi: 10.1090/S0002-9947-00-02483-1.  Google Scholar [20] D. C. Offin and H. Cabral, Hyperbolicity for symmetric periodic orbits in the isosceles three body problem,, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 379.  doi: 10.3934/dcdss.2009.2.379.  Google Scholar [21] G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem,, Ergod. Th. & Dynam. Sys., 27 (2007), 1947.   Google Scholar [22] C. Simó, Dynamical properties of the figure eight solution of the three body problem,, Contemp. Math., 292 (2002), 209.  doi: 10.1090/conm/292/04926.  Google Scholar [23] S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses,, Arch. Rational. Mech. Anal., 184 (2007), 465.  doi: 10.1007/s00205-006-0030-8.  Google Scholar
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