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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

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 & Continuous Dynamical Systems, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137
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show all references

References:
[1]

Funct. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.  Google Scholar

[2]

Encyclopaedia Math. Sci., 3, Springer-Verlag, Berlin, 1988.  Google Scholar

[3]

Funct. Anal. Appl., 19 (1985), 251-259.  Google Scholar

[4]

Celest. Mech. Dynam. Astron., 95 (2006), 55-66. 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]

Ann. Math., 152 (2000), 881-901. doi: 10.2307/2661357.  Google Scholar

[7]

Celest. Mech. Dynam. Astron., 77 (2000), 139-152. doi: 10.1023/A:1008381001328.  Google Scholar

[8]

Reviews in Math. Physics, 6 (1994), 1187-1232. doi: 10.1142/S0129055X94000432.  Google Scholar

[9]

Adv. Math., 21 (1976) 173-195. doi: 10.1016/0001-8708(76)90074-8.  Google Scholar

[10]

Springer-Verlag, New York, 1991. Google Scholar

[11]

Inv. Math., 155 (2004), 305-362. doi: 10.1007/s00222-003-0322-7.  Google Scholar

[12]

Amer. J. Math., 99 (1970), 961-971. doi: 10.2307/2373993.  Google Scholar

[13]

Comm. Math. Phys., 290 (2009), 737-777. doi: 10.1007/s00220-009-0860-y.  Google Scholar

[14]

Springer-Verlag, New York, 1991. Google Scholar

[15]

Celest. Mech. Dynam. Astron., 83 (2002), 325-353. doi: 10.1023/A:1020128408706.  Google Scholar

[16]

(Russian), MGU, Moscow, 1965. Google Scholar

[17]

J. Diff. Eqns., 41 (1981), 228-238. doi: 10.1016/0022-0396(81)90059-0.  Google Scholar

[18]

Celest. Mech. Dynam. Astron., 55 (1993), 289-303. doi: 10.1007/BF00692516.  Google Scholar

[19]

Trans. Amer. Math. Soc., 352 (2000), 3323-3338. doi: 10.1090/S0002-9947-00-02483-1.  Google Scholar

[20]

Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 379-392. doi: 10.3934/dcdss.2009.2.379.  Google Scholar

[21]

Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963.  Google Scholar

[22]

Contemp. Math., 292 (2002), 209-228. doi: 10.1090/conm/292/04926.  Google Scholar

[23]

Arch. Rational. Mech. Anal., 184 (2007), 465-493. doi: 10.1007/s00205-006-0030-8.  Google Scholar

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