Abstract
We consider the motion of a massless particle around an oblate planet, keeping only in the expression of the perturbing potential the second degree zonal harmonic. We prove the analytical non integrability of this problem, by using Ziglin's theorem and the Yoshida criterion for homogeneous potentials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Churchill, R.C. and Rod, D.L.: 1988, ‘Geometrical Aspects of Ziglin's Non-Integrability Theorem’,J. of Diff. Eq. 76, 91–114.
Hille, E.: 1976,Ordinary Differential Equations in the Complex Domain, J. Wiley and Sons, New York.
Ince, E.L.: 1956,Ordinary Differential Equations, Dover Publ., New York.
Ito, H.: 1985, ‘Non-integrability of Hénon-Heiles system and a Theorem of Ziglin’,Kodai Math. J. 8, 120–138.
Nahon, F.: ‘Le groupe de monodromie de l'équation de Hill et de l'équation de Gauss associée, unpublished notes, Univ. of Paris VI.
Plemelj, J.: 1964,Problem in the Sense of Riemann and Klein, Wiley-Interscience, New York.
Simó, C.: 1991, ‘Measuring the Lack of Integrability of theJ 2 Problem for Earth Satellites’, in Roy, A.E. (ed.),Predictability, Stability and Chaos in N-Body Dynamical Systems, Plenum Press, pp. 305–309.
Whittaker, E.T. and Watson, G.N.: 1927,A Course of Modern Analysis, Cambridge Univ. Press.
Yoshida, H.: 1987, ‘A Criterion for the Non-Existence of an Additional Integral in Hamiltonian Systems with a Homogeneous Potential’,Physica 29D, 128–142.
Yoshida, H., Ramani, A., and Grammaticos, B.: 1988, ‘Non-Integrability of the Fourth-Order Truncated Toda Hamiltonian’,Physica 30D, 151–163.
Ziglin, S.L.: 1983, ‘Branching of Solutions and Non-Existence of First Integrals in Hamiltonian Mechanics’,Funct. Anal. Appl. 16, 181–189, and17, 6–17.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Irigoyen, M., Simó, C. Non integrability of theJ 2 problem. Celestial Mech Dyn Astr 55, 281–287 (1993). https://doi.org/10.1007/BF00692515
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00692515