Abstract
We consider a Hamiltonian system with a one-parameter family of degenerate coisotropic invariant tori. We prove a theorem on the preservation of the majority of tori under small perturbations of the Hamiltonian.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. M. Samoilenko, Elements of the Mathematical Theory of Multifrequency Oscillations. Invariant Tori [in Russian], Nauka, Moscow (1987).
A. M. Samoilenko, “On some problems in the theory of perturbations of smooth invariant tori of dynamical systems,” Ukr. Mat. Zh., 46, No. 12, 1665–1699 (1994).
N. N. Nekhoroshev, “Action-angle variables and their generalizations,” Tr Mask. Mat Obshch., 26, 181–198 (1972).
J. J. Duistermaat, “On global action-angle coordinates,” Commun. Pure Appl. Math., 33, No. 5, 687–706 (1980).
I. O. Parasyuk, “Variables of the action-angle type on symplectic manifolds stratified by coisotropic tori,” Ukr. Mat. Zh., 45, No. 1, 77–85 (1993).
N. N. Bogolyubov, Yu. A. Mitropol’skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969)
V. I. Arnol’d, V. V. Kozlov, and A. I. Neishtadt. “Mathematical aspects of classical and celestial mechanics,” in: VINITI Series in Contemporary Problems of Mathematics [in Russian], Vol. 3, VINITI, Moscow (1985), pp 5–304
I. O. Parasyuk, “On the preservation of many-dimensional invariant tori of Hamiltonian systems,” Ukr Mat. Zh., 36, No. 4, 467–473 (1984).
V. I. Arnol’d, “Small denominators and problems of stability of motion in classical and celestial mechanics,” Usp Mat. Nauk, 18, No. 6, 91–192 (1963).
A. M. Samoilenko, “On the reducibility of a system of linear differential equations with quasiperiodic coefficients,” Ukr. Mat. Zh., 20, No. 2, 201–212 (1968).
A. N. Kolmogorov, “On the preservation of quasiperiodic motions under small variation of the Hamiltonian function,” Dokl. Akad. Nauk SSSR, 98, No. 4, 527–530 (1954).
Yu. Mozer, “On the expansion of quasiperiodic motions in convergent power series,” Usp Mat. Nauk, 24, No. 2, 165–217 (1969).
H. Rüssmann, “Non-degeneracy in the perturbation theory of integrable dynamical systems,” London Math. Soc. Lect. Notes Ser., No. 134, 15–18 (1989).
A. D. Bryuno, “On the conditions of nondegeneracy in the Kolmogorov theorem.” Dokl Ros Akad. Nauk, 322, No. 6, 1028–1032 (1992).
H. W. Broer, G. B. Huitema, and M. B. Sevryuk, Quasiperiodic Motions in Families of Dynamical Systems Order Admits Chaos, Preprint No. W-9519, Groningen University, Groningen (1995).
V. I. Arnol’d, “A Proof of the Kolmogorov theorem on the preservation of quasiperiodic motions under a small variation of the Hamiltonian function,” Usp. Mat. Nauk, 18, No. 5, 13–40 (1963).
H. Rüssmann, “On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus,” Lect. Notes Phys., 38, 598–624 (1975).
A. S. Pyartli, “Diophantine approximations on submanifolds of the Euclidean space,” Funkts. Anal. Prilozh., 3, No. 3, 59–62 (1969).
V. I. Bakhtin, “A strengthened external property of Chebyshev polynomials,” Moscow Univ. Math. Bull., 42, No. 2, 24–26 (1987).
S. M. Nikol’skii, A Course in Mathematical Analysis [in Russian], Vol. 2, Nauka, Moscow (1973).
Author information
Authors and Affiliations
Additional information
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 72–86, January, 1998.
Rights and permissions
About this article
Cite this article
Parasyuk, I.O. Perturbations of degenerate coisotropic invariant tori of Hamiltonian systems. Ukr Math J 50, 83–99 (1998). https://doi.org/10.1007/BF02514690
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02514690