Abstract
For the Hamiltonian systems of KAM type, it is proved that some lower dimensional invariant tori always exist in the resonance gaps although those maximum tori can not survive small perturbations in the generic case.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
[A1] Arnol'd, V.I.: “Mathematical Methods of Classical Mechanics”. Berlin, Heidelberg, New York: Springer, 1978
[A2] Arnol'd, V.I.: Proof of A.N. Kolmogorov's theorem on the preservation of quasi periodic motions under small perturbations of Hamiltonian. Usp. Math. USSR18, 9–36 (1963)
[Au] Aubry, S.: Theory of the devil's staircase. Lect. Notes Math.925 (1978–79)
[BK] Berstein, D., Katok, A.: Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians. Invent. Math.88, 225–241 (1987)
[Ch] Chierchia, L.: On the stability problem for nearly-integrable Hamiltonian systems. Seminar on Dynamical Systems (1994). Basel, Boston: Birkhäuser, pp. 35–46
[CZ] Conley, C.C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnol'd. Invent. Math.73, 33–49 (1983)
[E] Eliasson, L.H.: Perturbations of stable invariant tori for Hamiltonian systems. Ann. Sc. Norm. Super. Pisa. Cl. Sci. IV. Ser.15, 115–147 (1988)
[G] Graff, S.M.: On the continuation of hyperbolic invariant tori for Hamiltonian systems. J. Diff. Eqns.15, 1–69 (1974)
[H1] Herman, M.R.: Sur les courbes invariantes par les diffémorphisms de lánneau. Astérisque103 (1983)
[H2] Herman, M.R.: Sur les courbes invariantes par les diffémorphisms de lánneau. Astérisque144 (1986)
[K] Kolmogorov, A.N.: On quasi-periodic motions under small perturbations of the Hamiltonian. Doklady Akad. Nauk. USSR98, 1–20 (1954)
[L] Lochak, P.: Canonical perturbation theory via simultaneous approximation. Uspekhi Mat. Nauk47, 59–140 (1992)
[Ma1] Mather, J.N.: Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology21, 457–467 (1982)
[Ma2] Mather, J.N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z.207, 169–207 (1991)
[Mo1] Moser, J.: On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gött. Math-PhysK1, 1–20 (1962)
[Mo2] Moser, J.: convergent series expansions for quasi-periodic motions. Math. Ann.169, 136–176 (1967)
[MP] Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials. Comment. Math. Helvetici59, 39–85 (1984)
[P] Pöschel, J.: On elliptic lower dimensional tori in Hamiltonian systems. Math. Z.202, 559–608 (1989)
[Rob] Robinson, C.: Generic properties of conservative systems. Am. J. Math.92, 562–603, 897–906 (1970)
[Rüs1] Rüssmann, H.: On the existence of invariant curves of twist mappings of an annulus. Geometric Dynamics. Lect. Notes Math.1007, 677–718 (1983)
[Rüs2] Rüssmann, H.: On twist Hamiltonians. Talk on the Colloque International: Mécanique céleste et systemes Hamiltonian, Marseille
[SM] Siegel, C.L., Moser, J.: Lectures on Celestial Mechanics. Berlin, Heidelberg, New York: Springer, 1971
[T] Treshchev, D.V.: The mechanism of destruction of resonance tori of Hamiltonian systems. Math. USSR Sbornik68, 181–203 (1991)
[W] Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys.127, 479–528 (1990)
Author information
Authors and Affiliations
Additional information
Communicated by M. Herman
Rights and permissions
About this article
Cite this article
Cheng, CQ. Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems. Commun.Math. Phys. 177, 529–559 (1996). https://doi.org/10.1007/BF02099537
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02099537