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Discrete dynamics in implicit form
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 |
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. |
show all references
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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