Variational Criteria for Nonintegrability and Chaos in Hamiltonian Systems
Part of the
NATO ASI Series
book series (NSSB, volume 331)
The goal of this article is to show how existence results obtained by variational methods can be applied to prove chaotic behavior for a class of Hamiltonian systems. The standard tool is the perturbation theory, for example, the Poincaré-Melnikov-Arnold integral. This approach is restricted to the case when the system is in some sense close to integrable. If this is not so, non-perturbative methods are needed. Our method works best for natural analytic systems with two degrees of freedom. We obtain the existence of homoclinic orbits by variational methods and then prove, using analyticity, that they are isolated from purely topological reasons. Of course, it is impossible to prove transversality by variational methods. However, if the homoclinic orbit is obtained by minimizing the action functional, or by the mountain-pass theorem, then it is usually possible to prove a weaker property, that they are transversal in the topological sense. Then we can apply a modification of the well-known results on dynamics near transversal homoclinics.
KeywordsPeriodic Orbit Hamiltonian System Configuration Space Euler Characteristic Homoclinic Orbit
Bolotin, S.V., 1978, Libration orbits of natural dynamical systems, Vestnik Moskov. Univ. Ser. I Matem. Mekh.
Bolotin, S.V., 1984, Nonintegrability of the n
-center problem for n
> 2, Vestnik Moskov. Univ. Ser. I Matem. Mekh.
Bolotin, S.V., 1984, Influence of singularities of the potential energy on the integrability of dynamical systems, Prikl. Matem. i Mekhan.
Bolotin, S.V., 1992, Variational methods for constructing chaotic motions in the rigid body dynamics, Prikl. Matem. i Mekhan.
Bolotin, S.V., 1992, Doubly asymptotic orbits of minimal geodesies, Vestnik Moskov. Univ. Ser. I Matem. Mekh.
Coti-Zelati, V., Ekeland, I., and Séré, E., 1990, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann.
Coti-Zelati, V., and Rabinowitz, P., 1991, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, Jour. Amer. Math. Soc.
Devaney, R.L., 1978, Transversal homoclinic orbits in an integrable system, Amer. J. Math.
Katok, A., 1982, Entropy and closed geodesies, Ergod. Th. and Dynam. Syst.
Kozlov, V.V., 1979, Topological obstructions to the integrability of natural mechanical systems, Dokl. Akad. Nauk. SSSR.
Kozlov, V.V., 1983, Integrability and nonintegrability in classical mechanics, Uspekhi Mat. Nauk.
Paternain, G.P., 1992, On the topology of manifolds with completely integrable geodesic flows, Ergod. Th. and Dynam. Sys.
Séré, E., 1992, Looking for the Bernoulli shift, Preprint
, Universite Paris Dauphine.Google Scholar
Taymanov, I.A., 1988, On topological properties of integrable geodesic flows, Mat. Zametki.
Turayev, D.V., and Shilnikov, L.P., 1989, On Hamiltonian systems with homoclinic curves of a saddle, Dokl AN SSSR
© Springer Science+Business Media New York 1994