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Non-integrability criterium for normal variational equations around an integrable subsystem and an example: The Wilberforce spring-pendulum
Horseshoe periodic orbits with one symmetry in the general planar three-body problem
1. | Departamento de Matemáticas, Facultad de Ciencias, UNAM, Ciudad Universitaria, México, D.F. 04510, Mexico |
2. | Department of Mathematics, Facultad de Ciencias, UNAM, Ciudad Universitaria, México, D.F. 04510 |
3. | Departamento de Matemáticas, Universidad Autónoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186, México, D.F. 09340 |
References:
[1] |
E. Barrabés and S. Mikkola, Families of periodic horseshoe orbits in the restricted three-body problem, Astron, Astrophys, 432 (2005), 1115-1129.
doi: 10.1051/0004-6361:20041483. |
[2] |
A. Bengochea and E. Piña, The Saturn, Janus and Epimetheus dynamics as a gravitational three-body problem in the plane, Rev. Mexicana Fís., 55 (2009), 97-105. |
[3] |
A. Bengochea, M. Falconi and E. Pérez-Chavela, Symmetric horseshoe periodic orbits in the general planar three-body problem, Astrophys. Space Sci., 333 (2011), 399-408.
doi: 10.1007/s10509-011-0641-x. |
[4] |
J. M. Cors and G. R. Hall, Coorbital periodic orbits in the three body problem, SIAM J. Appl. Dyn. Syst., 2 (2003), 219-237.
doi: 10.1137/S1111111102411304. |
[5] |
S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. I. Theory, Icarus, 48 (1981), 1-11.
doi: 10.1016/0019-1035(81)90147-0. |
[6] |
S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. II. The coorbital satellites of Saturn, Icarus, 48 (1981), 12-22.
doi: 10.1016/0019-1035(81)90148-2. |
[7] |
J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26.
doi: 10.1016/0771-050X(80)90013-3. |
[8] |
M. Hénon and J. M. Petit, Series expansion of encounter-type solutions of Hill's problem, Celest. Mech. Dynam. Astron., 38 (1986), 67-100. |
[9] |
X. Y. Hou and L. Liu, The symmetric horseshoe periodic families and the lyapunov planar family around $L_3$, Astron. J., 136 (2008), 67-75.
doi: 10.1088/0004-6256/136/1/67. |
[10] |
J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Phys. D, 112 (1998), 1-39.
doi: 10.1016/S0167-2789(97)00199-1. |
[11] |
J. Llibre and M. Ollé, The motion of Saturn coorbital satellites in the restricted three-body problem, Astron. Astrophys, 378 (2001), 1087-1099.
doi: 10.1051/0004-6361:20011274. |
[12] |
K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'' $1^{st}$ edition, Springer-Verlag, New York, 1992. |
[13] |
F. J. Muñoz-Almaraz, J. Galán and E. Freire, Families of symmetric periodic orbits in the three body problem and the figure eight, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 25 (2004), 229-240. |
[14] |
J. M. Petit and M. Hénon, Satellite encounters, Icarus, 66 (1986), 536-555.
doi: 10.1016/0019-1035(86)90089-8. |
[15] |
A. E. Roy and M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system. II. The mirror theorem, Mon. Not. R. Astron. Soc., 115 (1955), 296-309. |
[16] |
F. Spirig and J. Waldvogel, The three-body problem with two small masses: A singular-perturbation approach to the problem of Saturn's coorbiting satellites, in "Stability of the Solar System and its Minor Natural and Artificial Bodies,'' (ed. V. G. Szebehely), Reidel, (1985), 53-63.
doi: 10.1007/978-94-009-5398-7_5. |
[17] |
C. F. Yoder, G. Colombo, S. P. Synnott and K. A. Yoder, Theory of motion of Saturn's coorbiting satellites, Icarus, 53 (1983), 431-443.
doi: 10.1016/0019-1035(83)90207-5. |
[18] |
C. F. Yoder, S. P. Synnott and H. Salo, Orbits and masses of Saturn's co-orbiting satellites, Janus and Epimetheus, Astron. J., 98 (1989), 1875-1889.
doi: 10.1086/115265. |
[19] |
J. Waldvogel and F. Spirig, Co-orbital satellites and hill's lunar problem, in "Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems'' (ed. A. E. Roy), Kluwer, (1988), 223-234.
doi: 10.1007/978-94-009-3053-7_20. |
show all references
References:
[1] |
E. Barrabés and S. Mikkola, Families of periodic horseshoe orbits in the restricted three-body problem, Astron, Astrophys, 432 (2005), 1115-1129.
doi: 10.1051/0004-6361:20041483. |
[2] |
A. Bengochea and E. Piña, The Saturn, Janus and Epimetheus dynamics as a gravitational three-body problem in the plane, Rev. Mexicana Fís., 55 (2009), 97-105. |
[3] |
A. Bengochea, M. Falconi and E. Pérez-Chavela, Symmetric horseshoe periodic orbits in the general planar three-body problem, Astrophys. Space Sci., 333 (2011), 399-408.
doi: 10.1007/s10509-011-0641-x. |
[4] |
J. M. Cors and G. R. Hall, Coorbital periodic orbits in the three body problem, SIAM J. Appl. Dyn. Syst., 2 (2003), 219-237.
doi: 10.1137/S1111111102411304. |
[5] |
S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. I. Theory, Icarus, 48 (1981), 1-11.
doi: 10.1016/0019-1035(81)90147-0. |
[6] |
S. F. Dermott and C. D. Murray, The dynamics of tadpole and horseshoe orbits. II. The coorbital satellites of Saturn, Icarus, 48 (1981), 12-22.
doi: 10.1016/0019-1035(81)90148-2. |
[7] |
J. R. Dormand and P. J. Prince, A family of embedded Runge-Kutta formulae, J. Comput. Appl. Math., 6 (1980), 19-26.
doi: 10.1016/0771-050X(80)90013-3. |
[8] |
M. Hénon and J. M. Petit, Series expansion of encounter-type solutions of Hill's problem, Celest. Mech. Dynam. Astron., 38 (1986), 67-100. |
[9] |
X. Y. Hou and L. Liu, The symmetric horseshoe periodic families and the lyapunov planar family around $L_3$, Astron. J., 136 (2008), 67-75.
doi: 10.1088/0004-6256/136/1/67. |
[10] |
J. S. W. Lamb and J. A. G. Roberts, Time-reversal symmetry in dynamical systems: A survey, Phys. D, 112 (1998), 1-39.
doi: 10.1016/S0167-2789(97)00199-1. |
[11] |
J. Llibre and M. Ollé, The motion of Saturn coorbital satellites in the restricted three-body problem, Astron. Astrophys, 378 (2001), 1087-1099.
doi: 10.1051/0004-6361:20011274. |
[12] |
K. R. Meyer and G. R. Hall, "Introduction to Hamiltonian Dynamical Systems and the N-Body Problem,'' $1^{st}$ edition, Springer-Verlag, New York, 1992. |
[13] |
F. J. Muñoz-Almaraz, J. Galán and E. Freire, Families of symmetric periodic orbits in the three body problem and the figure eight, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 25 (2004), 229-240. |
[14] |
J. M. Petit and M. Hénon, Satellite encounters, Icarus, 66 (1986), 536-555.
doi: 10.1016/0019-1035(86)90089-8. |
[15] |
A. E. Roy and M. W. Ovenden, On the occurrence of commensurable mean motions in the solar system. II. The mirror theorem, Mon. Not. R. Astron. Soc., 115 (1955), 296-309. |
[16] |
F. Spirig and J. Waldvogel, The three-body problem with two small masses: A singular-perturbation approach to the problem of Saturn's coorbiting satellites, in "Stability of the Solar System and its Minor Natural and Artificial Bodies,'' (ed. V. G. Szebehely), Reidel, (1985), 53-63.
doi: 10.1007/978-94-009-5398-7_5. |
[17] |
C. F. Yoder, G. Colombo, S. P. Synnott and K. A. Yoder, Theory of motion of Saturn's coorbiting satellites, Icarus, 53 (1983), 431-443.
doi: 10.1016/0019-1035(83)90207-5. |
[18] |
C. F. Yoder, S. P. Synnott and H. Salo, Orbits and masses of Saturn's co-orbiting satellites, Janus and Epimetheus, Astron. J., 98 (1989), 1875-1889.
doi: 10.1086/115265. |
[19] |
J. Waldvogel and F. Spirig, Co-orbital satellites and hill's lunar problem, in "Long-Term Dynamical Behaviour of Natural and Artificial N-Body Systems'' (ed. A. E. Roy), Kluwer, (1988), 223-234.
doi: 10.1007/978-94-009-3053-7_20. |
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