American Institute of Mathematical Sciences

August  2015, 35(8): 3707-3719. doi: 10.3934/dcds.2015.35.3707

Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities

 1 Division of Mathematical Science, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan

Received  April 2014 Revised  December 2014 Published  February 2015

It is a big problem to distinguish between integrable and non-integrable Hamiltonian systems. We provide a new approach to prove the non-integrability of homogeneous Hamiltonian systems with two degrees of freedom. The homogeneous degree can be taken from real values (not necessarily integer). The proof is based on the blowing-up theory which McGehee established in the collinear three-body problem. We also compare our result with Molares-Ramis theory which is the strongest theory in this field.
Citation: Mitsuru Shibayama. Non-integrability criterion for homogeneous Hamiltonian systems via blowing-up technique of singularities. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3707-3719. doi: 10.3934/dcds.2015.35.3707
References:
 [1] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Springer, New York, 1989. doi: 10.1007/978-1-4757-2063-1. [2] H. Bruns, Über die Integrale des Vielkörper-Problems, Acta Math., 11 (1887), 25-96. doi: 10.1007/BF02612319. [3] R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017. [4] R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem, Celestial Mech., 28 (1982), 25-36. doi: 10.1007/BF01230657. [5] G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615. doi: 10.3934/dcds.2014.34.4589. [6] S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. doi: 10.1007/BF02592182. [7] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175. [8] R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal., 15 (1984), 857-876. doi: 10.1137/0515065. [9] J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser Basel, 1999. doi: 10.1007/978-3-0348-8718-2. [10] J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120. [11] H. Poincaré, New Methods of Celestial Mechanics Vol. 1, American Institute of Physics, 1993. [12] M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen., 30 (1997), 5869-5876. doi: 10.1088/0305-4470/30/16/026. [13] M. Shibayama, Non-integrability of the collinear three-body problem, Discrete Contin. Dyn. Syst., 30 (2011), 299-312. doi: 10.3934/dcds.2011.30.299. [14] M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403. doi: 10.1088/0951-7715/22/10/004. [15] H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems, Physica, 21 (1986), 163-170. doi: 10.1016/0167-2789(86)90087-4. [16] H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9. [17] M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians, RIMS Kokyuroku Bessatsu, B40 (2013), 177-189. [18] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I. Funktsional. Anal. i Prilozhen., 16 (1982), 30-41, 96. [19] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II. Funktsional. Anal. i Prilozhen., 17 (1983), 8-23.

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References:
 [1] V. I. Arnol'd, Mathematical Methods of Classical Mechanics, $2^{nd}$ edition, Springer, New York, 1989. doi: 10.1007/978-1-4757-2063-1. [2] H. Bruns, Über die Integrale des Vielkörper-Problems, Acta Math., 11 (1887), 25-96. doi: 10.1007/BF02612319. [3] R. L. Devaney, Triple collision in the planar isosceles three-body problem, Invent. Math., 60 (1980), 249-267. doi: 10.1007/BF01390017. [4] R. L. Devaney, Motion near total collapse in the planar isosceles three-body problem, Celestial Mech., 28 (1982), 25-36. doi: 10.1007/BF01230657. [5] G. Duval and A. J. Maciejewski, Integrability of Hamiltonian systems with homogeneous potentials of degrees 2. An application of higher order variational equations, Discrete Contin. Dyn. Syst., 34 (2014), 4589-4615. doi: 10.3934/dcds.2014.34.4589. [6] S. Kovalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. doi: 10.1007/BF02592182. [7] R. McGehee, Triple collision in the collinear three-body problem, Invent. Math., 27 (1974), 191-227. doi: 10.1007/BF01390175. [8] R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM J. Math. Anal., 15 (1984), 857-876. doi: 10.1137/0515065. [9] J. J. Morales-Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Birkhaeuser Basel, 1999. doi: 10.1007/978-3-0348-8718-2. [10] J. J. Morales-Ruiz and J. P. Ramis, A note on the non-integrability of some Hamiltonian systems with a homogeneous potential, Methods Appl. Anal., 8 (2001), 113-120. [11] H. Poincaré, New Methods of Celestial Mechanics Vol. 1, American Institute of Physics, 1993. [12] M. E. Sansaturio, I. Vigo-Aguiar and J. M. Ferrándiz, Non-integrability of some Hamiltonian systems in polar coordinates, J. Phys. A: Math. Gen., 30 (1997), 5869-5876. doi: 10.1088/0305-4470/30/16/026. [13] M. Shibayama, Non-integrability of the collinear three-body problem, Discrete Contin. Dyn. Syst., 30 (2011), 299-312. doi: 10.3934/dcds.2011.30.299. [14] M. Shibayama and K. Yagasaki, Heteroclinic connections between triple collisions and relative periodic orbits in the isosceles three-body problem, Nonlinearity, 22 (2009), 2377-2403. doi: 10.1088/0951-7715/22/10/004. [15] H. Yoshida, Existence of exponentially unstable periodic solutions and the nonintegrability of homogeneous Hamiltonian systems, Physica, 21 (1986), 163-170. doi: 10.1016/0167-2789(86)90087-4. [16] H. Yoshida, A criterion for the nonexistence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9. [17] M. Yoshino, Smooth-integrable and analytic-nonintegrable resonant Hamiltonians, RIMS Kokyuroku Bessatsu, B40 (2013), 177-189. [18] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I. Funktsional. Anal. i Prilozhen., 16 (1982), 30-41, 96. [19] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. II. Funktsional. Anal. i Prilozhen., 17 (1983), 8-23.
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