A non-integrability test for perturbed separable planar Hamiltonians. (English) Zbl 1115.37334
Summary: We prove the non-integrability of perturbed separable planar potentials by using the perturbed normal variational equations, with a reasoning analogous to Ziglin’s theorem. We apply the above criterion to a Hamiltonian that cannot be proved non-integrable by other known non-integrability criteria.
MSC:
| 37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 70H07 | Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics |
References:
| [1] | H. Poincaré, 1892, Les Méthodes Nouvelles de la Méchanique Céleste, vol. I, Gauthier-Villars, Paris. English translation: Goroff D.L. (Ed.), New Methods in Celestial Mechanics, American Institute of Physics, 1993.; H. Poincaré, 1892, Les Méthodes Nouvelles de la Méchanique Céleste, vol. I, Gauthier-Villars, Paris. English translation: Goroff D.L. (Ed.), New Methods in Celestial Mechanics, American Institute of Physics, 1993. |
| [2] | Ziglin, S. L., Func. Anal. Appl., 16, 181 (1983) |
| [3] | Ziglin, S. L., Func. Anal. Appl., 17, 6 (1983) · Zbl 0518.58016 |
| [4] | Yoshida, H., Physica D, 29, 128 (1987) · Zbl 0659.70012 |
| [5] | Moralez-Ruiz, J. J.; Simó, C., J. Diff. Eq., 107, 140 (1994) · Zbl 0799.58035 |
| [6] | J.J. Morales-Ruiz, C. Ramis, 1998a, Galoisian obstructions to integrability of Hamiltonian systems (preprint).; J.J. Morales-Ruiz, C. Ramis, 1998a, Galoisian obstructions to integrability of Hamiltonian systems (preprint). · Zbl 0982.37061 |
| [7] | J.J. Morales-Ruiz, C. Ramis, 1998b, Galoisian obstructions to integrability of Hamiltonian systems II (preprint).; J.J. Morales-Ruiz, C. Ramis, 1998b, Galoisian obstructions to integrability of Hamiltonian systems II (preprint). · Zbl 1140.37354 |
| [8] | Ito, H., Kodai Math. J., 8, 120 (1995) |
| [9] | Ito, H., ZAMP, 38, 459 (1987) |
| [10] | Ito, H., Bol. Soc. Bras. Mat., 21, 95 (1990) · Zbl 0762.58015 |
| [11] | Ichtiaroglou, S., J. Phys. A: Math. Gen., 22, 3461 (1989) · Zbl 0702.58026 |
| [12] | Almeida, M. A.; Moreira, I. C.; Yoshida, H., J. Phys. A: Math. Gen., 25, L227 (1992) · Zbl 0753.35103 |
| [13] | Irigoyen, M.; Simó, C., Celest. Mech. Dyn. Astr., 55, 281 (1993) · Zbl 0767.70015 |
| [14] | Françoise, J.-P.; Irigoyen, M., Geom. Phys., 10, 231 (1993) · Zbl 0779.58016 |
| [15] | Ferrándiz, J. M.; Sansaturio, M. E., Phys. Lett. A, 207, 180 (1995) · Zbl 1020.70502 |
| [16] | Ferrándiz, J. M.; Sansaturio, M. E.; Vigo, I., Phys. Lett. A, 221, 153 (1996) · Zbl 1098.70541 |
| [17] | Sansaturio, M. E.; Vigo-Aguiar, I.; Ferrándiz, J. M., J. Phys. A: Math. Gen., 30, 5869 (1997) · Zbl 0916.58016 |
| [18] | M.E. Sansaturio, I. Vigo-Aguiar, J.M. Ferrándiz, in: B.A. Steves, A.E. Roy (Eds.), The Dynamics of Small Bodies in the Solar System, Kluwer, the Netherlands, 1999, pp. 295-302.; M.E. Sansaturio, I. Vigo-Aguiar, J.M. Ferrándiz, in: B.A. Steves, A.E. Roy (Eds.), The Dynamics of Small Bodies in the Solar System, Kluwer, the Netherlands, 1999, pp. 295-302. |
| [19] | M.I. Vigo-Aguiar, 1999, No integrabilidad del problema del satélite, Doctor’s Thesis, Universidad de Alicante.; M.I. Vigo-Aguiar, 1999, No integrabilidad del problema del satélite, Doctor’s Thesis, Universidad de Alicante. |
| [20] | Churchill, R. C.; Rod, D. L., J. Diff. Eq., 76, 91 (1988) · Zbl 0661.58013 |
| [21] | Churchill, R. C.; Rod, D. L., SIAM J. Math. Annal., 2, 1790 (1991) |
| [22] | Meletlidou, E.; Ichtiaroglou, S., Physica D, 71, 261 (1994) · Zbl 0804.58025 |
| [23] | Meletlidou, E.; Ichtiaroglou, S., J. Phys. A, 27, 3919 (1994) · Zbl 0842.70008 |
| [24] | Meletlidou, E.; Ichtiaroglou, S., Phys. Lett. A, 188, 157 (1994) · Zbl 0941.70510 |
| [25] | Mel’nikov, V. K., Trans. Moscow Math. Soc., 12, 3 (1963) · Zbl 0135.31001 |
| [26] | Kozlov, V. V., J. Appl. Math. Mech., 42, 420 (1979) |
| [27] | Yoshida, H., Commun. Math. Phys., 116, 529 (1988) · Zbl 0671.35074 |
| [28] | G.D. Birkhoff, 1927, Dynamical Systems, Colloquium Publications, vol. IX, Am. Math. Soc., Providence, Rhode Island, Acta Math. 50.; G.D. Birkhoff, 1927, Dynamical Systems, Colloquium Publications, vol. IX, Am. Math. Soc., Providence, Rhode Island, Acta Math. 50. · JFM 53.0732.01 |
| [29] | V.V. Kozlov, 1996, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, Berlin.; V.V. Kozlov, 1996, Symmetries, Topology and Resonances in Hamiltonian Mechanics, Springer, Berlin. |
| [30] | K.R. Meyer, G.R. Hall, 1992, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York.; K.R. Meyer, G.R. Hall, 1992, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer, New York. · Zbl 0743.70006 |
| [31] | H.T. Davis, 1962, Introduction to Nonlinear Differential and Integral Equations, Dover, New York.; H.T. Davis, 1962, Introduction to Nonlinear Differential and Integral Equations, Dover, New York. · Zbl 0106.28904 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
