Abstract
We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
I. O. Parasyuk, “Bifurcation of the Cantor set of coisotropic invariant tori of a Hamiltonian system under perturbations of a symplectic structure,” Nelin. Kolyv. 1, No. 2, 81-89 (1998).
J. Pöschel, “Integrability of Hamiltonian systems on Cantor sets,” Commun. Pure Appl. Math. 35, No. 5, 653-695 (1982).
V. I. Arnol'd, Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989).
V. W. Guillemin and S. Sternberg, Geometrical Asymptotics AMS, Providence, R.I. (1977).
J. Moser, “On the volume elements on a manifold,” Trans. Amer. Math. Soc. 120, 286-294 (1965).
H. Rüssmann, “Non-degeneracy in the perturbation theory of integrable dynamical systems,” London Math. Soc. Lect. Notes Ser. No. 134, 15-18 (1989).
T. Kasuga, “On the adiabatic theorem for the Hamiltonian system of differential equations in the classical mechanics,” Proc. Jpn. Acad. 37, 366-382 (1961).
A. I. Neishtadt, “On averaging in multifrequency systems,” Dokl. Akad. Nauk SSSR 226, No. 6, 1295-1298 (1976).
A. M. Samoilenko and R. I. Petrishin, “On integral manifolds of multifrequency oscillation systems,” Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 2, 378-395 (1990).
A. M. Samoilenko, “N. N. Bogolyubov and nonlinear mechanics,” Usp. Mat. Nauk 49, No. 5, 103-146 (1994).
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. Math. Nauk 18, No. 5, 13-40 (1963).
J. Moser, “Convergent series expansions for quasiperiodic motions,” Math. Ann. 169, 136-176 (1967).
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. A. Kubichka and I. O. Parasyuk, “Whitney-smooth family of coisotropic invariant tori of a Hamiltonian system close to a degenerate system,” Visn. Kyiv. Univ., Ser. Mat. Mekh. Issue 4, 20-29 (2000).
N. N. Bogolyubov, Yu. A. Mitropol'skii, and A. M. Samoilenko, Method of Accelerated Convergence in Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1969).
H. W. Broer, G. B. Huitema, and M. B. Sevryuk, “Quasiperiodic motions in families of dynamical systems: order admits chaos,” Lect. Notes Math. 1645, 1-195 (1997).
M. B. Sevryuk, “Invariant tori of Hamiltonian systems that are nondegenerate in Rüessman's sense,” Dokl. Akad. Nauk Rossii 346, No. 5, 590-593 (1996).
I. O. Parasyuk, “Perturbations of degenerate coisotropic invariant tori of Hamiltonian systems,” Ukr. Mat. Zh. 50, No. 1, 72-86 (1998).
A. S. Pyartli, “Diophantine approximations on submanifolds of the Euclidean space,” Funkts. Anal. Prilozhen. 3, No. 3, 59-62 (1969).
V. I. Bakhtin, “A strengthened extremal property of Chebyshev polynomials,” Moscow Univ. Math. Bull. 42, No. 2, 24-26 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kubichka, A.A., Parasyuk, I.O. Bifurcation of a Whitney-Smooth Family of Coisotropic Invariant Tori of a Hamiltonian System under Small Deformations of a Symplectic Structure. Ukrainian Mathematical Journal 53, 701–718 (2001). https://doi.org/10.1023/A:1012574132151
Issue Date:
DOI: https://doi.org/10.1023/A:1012574132151