×

Perturbations of stable invariant tori for Hamiltonian systems. (English) Zbl 0685.58024

It is well known that the stable invariant tori of maximal dimension of an integrable Hamiltonian system in general persist under perturbations of the Hamiltonian. In this paper the perturbation theory of the lower dimensional stable tori is studied, and it is shown that under an appropriate non-degeneracy condition on the Hamiltonian many of these tori persist under perturbations.
Reviewer: Ch.Bär

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] V.I. Arnold , Mathematical methods of classical mechanics , Springer ( 1978 ). MR 690288 | Zbl 0386.70001 · Zbl 0386.70001
[2] J. Vey , Sur certains systèmes dynamiques séparables , American Journal of Mathematics , 100 ( 1978 ), pp. 591 - 614 . MR 501141 | Zbl 0384.58012 · Zbl 0384.58012
[3] A.N. Kolmogorov , The general theory of dynamical systems and classical mechanics, Proc. of the 1954 Intern. Congr. of Math. , in R. Abraham, J.E. Marsden, Foundations of mechanics , Benjamin ( 1978 ). · Zbl 0056.31502
[4] V.I. Arnold , Proof of a theorem of A.N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian , Russian Mathematical Surveys , 18 ( 1962 ), No. 5 , pp. 9 - 36 . MR 163025 | Zbl 0129.16606 · Zbl 0129.16606
[5] J. Moser , On the theory of quasiperiodic motions , Siam Review 8 ( 1966 ), pp. 145 - 172 . MR 203160 | Zbl 0243.34081 · Zbl 0243.34081
[6] J. Moser , Convergent series expansions for quasiperiodic motions , Mathematische Annalen , 169 ( 1967 ), pp. 136 - 176 . MR 208078 | Zbl 0149.29903 · Zbl 0149.29903
[7] H. Poincaré , Les méthodes nouvelles de la mécanique céleste , vol. I , Paris ( 1892 ). JFM 30.0834.08 · JFM 30.0834.08
[8] S. Graff , On the conservation of hyperbolic invariant tori for Hamiltonian systems , Journal of Differential Equations , 15 ( 1974 ), pp. 1 - 69 . MR 365626 | Zbl 0257.34048 · Zbl 0257.34048
[9] E. Zehnder , Generalized implicit function theorems with application to some small divisor problems II , Communications on Pure and Applied Mathematics , 29 ( 1976 ), pp. 49 - 111 . MR 426055 | Zbl 0334.58009 · Zbl 0334.58009
[10] V.K. Melnikov , On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function , Soviet Mathematics Doklady , 6 ( 1965 ), pp. 1592 - 1596 . Zbl 0143.11801 · Zbl 0143.11801
[11] V.K. Melnikov , A family of conditionally periodic solutions of a Hamiltonian system , Soviet Mathematics Doklady , 9 ( 1968 ), No. 4 , pp. 882 - 885 . Zbl 0185.17101 · Zbl 0185.17101
[12] J. Moser - J. Pöschel , An extension of a result by Dinaburg and Sinai on quasiperiodic potentials , Commentarii Mathematici Helvetici , 59 ( 1984 ), pp. 39 - 85 . MR 743943 | Zbl 0533.34023 · Zbl 0533.34023
[13] C.L. Charlier , Die Mechanik des Himmels , vol. 2 , Leipzig ( 1907 ). JFM 38.0949.11 · JFM 38.0949.11
[14] A.S. Pyartli , Diophantine approximation on submanifolds of euclidean space , Functional Analysis and its Applications , 3 ( 1969 ), pp. 303 - 306 . Zbl 0216.04401 · Zbl 0216.04401
[15] H. Rüssmann , On optimal estimates for the solutions of linear partial differential equations of the first order with constant coefficients on the torus , Springer, Lecture Notes in Physics , Vol. 38 ( 1975 ), pp. 598 - 624 . MR 467824 | Zbl 0319.35017 · Zbl 0319.35017
[16] C.L. Siegel - J. Moser , Lectures on celestial mechanics , Springer ( 1971 ). MR 502448 | Zbl 0312.70017 · Zbl 0312.70017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.