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

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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

Mark Lewis ^1, , Daniel Offin ^1, , Pietro-Luciano Buono ^2, and Mitchell Kovacic ^2,

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

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

Mathematics Subject Classification: 34C35, 34C27, 54H2.

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 and Continuous Dynamical Systems, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137

References:

[1]	V. I. Arnol'd, Characteristic class entering in quantization conditions, Funct. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.
[2]	V. I. Arnol'd, "Dynamical Systems III," Encyclopaedia Math. Sci., 3, Springer-Verlag, Berlin, 1988.
[3]	V. I. Arnol'd, Sturm theorems and symplectic geometry, Funct. Anal. Appl., 19 (1985), 251-259.
[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), 55-66. doi: 10.1007/s10569-006-9016-y.
[5]	P. L. Buono, M. Kovacic, M. Lewis and D. Offin, Symmetry-breaking bifurcations of the hip-hop orbit,, in preparation., ().
[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), 881-901. doi: 10.2307/2661357.
[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), 139-152. doi: 10.1023/A:1008381001328.
[8]	G. F. Dell'Antonio, Variational calculus and stability of periodic solutions of a class of Hamiltonian systems, Reviews in Math. Physics, 6 (1994), 1187-1232. doi: 10.1142/S0129055X94000432.
[9]	J. J. Duistermaat, On the morse index in variational calculus, Adv. Math., 21 (1976) 173-195. doi: 10.1016/0001-8708(76)90074-8.
[10]	I. Ekeland, "Convexity Methods in Hamiltonian Systems," Springer-Verlag, New York, 1991.
[11]	D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.
[12]	W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1970), 961-971. doi: 10.2307/2373993.
[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), 737-777. doi: 10.1007/s00220-009-0860-y.
[14]	J. Marsden, "Lectures on Mechanics," Springer-Verlag, New York, 1991.
[15]	C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dynam. Astron., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.
[16]	V. P. Maslov, "Theory of Perturbations and Asymptotic Methods," (Russian), MGU, Moscow, 1965.
[17]	K. R. Meyer, Hamiltonian systems with a discrete symmetry, J. Diff. Eqns., 41 (1981), 228-238. doi: 10.1016/0022-0396(81)90059-0.
[18]	K. R. Meyer and D. S. Schmidt, Librations of central configurations and braided Saturn rings, Celest. Mech. Dynam. Astron., 55 (1993), 289-303. doi: 10.1007/BF00692516.
[19]	D. C. Offin, Hyperbolic minimizing geodesics, Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1.
[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), 379-392. doi: 10.3934/dcdss.2009.2.379.
[21]	G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963.
[22]	C. Simó, Dynamical properties of the figure eight solution of the three body problem, Contemp. Math., 292 (2002), 209-228. doi: 10.1090/conm/292/04926.
[23]	S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses, Arch. Rational. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.

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References:

[1]	V. I. Arnol'd, Characteristic class entering in quantization conditions, Funct. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.
[2]	V. I. Arnol'd, "Dynamical Systems III," Encyclopaedia Math. Sci., 3, Springer-Verlag, Berlin, 1988.
[3]	V. I. Arnol'd, Sturm theorems and symplectic geometry, Funct. Anal. Appl., 19 (1985), 251-259.
[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), 55-66. doi: 10.1007/s10569-006-9016-y.
[5]	P. L. Buono, M. Kovacic, M. Lewis and D. Offin, Symmetry-breaking bifurcations of the hip-hop orbit,, in preparation., ().
[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), 881-901. doi: 10.2307/2661357.
[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), 139-152. doi: 10.1023/A:1008381001328.
[8]	G. F. Dell'Antonio, Variational calculus and stability of periodic solutions of a class of Hamiltonian systems, Reviews in Math. Physics, 6 (1994), 1187-1232. doi: 10.1142/S0129055X94000432.
[9]	J. J. Duistermaat, On the morse index in variational calculus, Adv. Math., 21 (1976) 173-195. doi: 10.1016/0001-8708(76)90074-8.
[10]	I. Ekeland, "Convexity Methods in Hamiltonian Systems," Springer-Verlag, New York, 1991.
[11]	D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical $n$-body problem, Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.
[12]	W. B. Gordon, A minimizing property of Keplerian orbits, Amer. J. Math., 99 (1970), 961-971. doi: 10.2307/2373993.
[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), 737-777. doi: 10.1007/s00220-009-0860-y.
[14]	J. Marsden, "Lectures on Mechanics," Springer-Verlag, New York, 1991.
[15]	C. Marchal, How the method of minimization of action avoids singularities, Celest. Mech. Dynam. Astron., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.
[16]	V. P. Maslov, "Theory of Perturbations and Asymptotic Methods," (Russian), MGU, Moscow, 1965.
[17]	K. R. Meyer, Hamiltonian systems with a discrete symmetry, J. Diff. Eqns., 41 (1981), 228-238. doi: 10.1016/0022-0396(81)90059-0.
[18]	K. R. Meyer and D. S. Schmidt, Librations of central configurations and braided Saturn rings, Celest. Mech. Dynam. Astron., 55 (1993), 289-303. doi: 10.1007/BF00692516.
[19]	D. C. Offin, Hyperbolic minimizing geodesics, Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1.
[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), 379-392. doi: 10.3934/dcdss.2009.2.379.
[21]	G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963.
[22]	C. Simó, Dynamical properties of the figure eight solution of the three body problem, Contemp. Math., 292 (2002), 209-228. doi: 10.1090/conm/292/04926.
[23]	S. Terracini and A. Venturelli, Symmetric trajectories for the 2N-body problem with equal masses, Arch. Rational. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.

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