Summary
In this paper a proof is given of Kolmogorov’s theorem on the existence of invariant tori in nearly integrable Hamiltonian systems. The scheme of proof is that of Kolmogorov, the only difference being in the way canonical transformations near the identity are defined. Precisely, use is made of the Lie method, which avoids any inversion and thus any use of the implicit-function theorem. This technical fact eliminates a spurious ingredient and simplifies the establishment of a central estimate.
Riassunto
Nel presente lavoro si dimostra il teorema di Kolmogorov sull’esistenza di tori invarianti in sistemi Hamiltoniani quasi integrabili. Si usa lo schema di dimostrazione di Kolmogorov, con la sola variante del modo in cui si definiscono le trasformazioni canoniche prossime all’identità. Si usa infatti il metodo di Lie, che elimina la necessità d’inversioni e quindi dell’impiego del teorema delle funzioni implicite. Questo fatto tecnico evita un ingrediente spurio e semplifica il modo in cui si ottiene una delle stime principali.
Резюме
В этой работе предлагается доказательство теоремы Колмогорова о существовании инвариантных торов в квази-интегрируемых Гамильтоновых системах. Используется схема доказательства, предложенная Колмогоровым, единственное отличие состоит в способе, которым определяются канонические преобразования. В этой работе используется метод Ли, которыи исключает необходимость инверсии и, следовательно, использование теоремы для неявной функции. Этот технический прием исключает ложный ингрдеиент и упрощает получение главной оценки.
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References
A. N. Kolmogorov:Dokl. Akad. Nauk SSSR,98, 527 (1954) (Math. Rev.,16, No. 924); English translation inG. Casati andJ. Ford (Editors):Lecture Notes in Physics, No. 93 (Berlin, 1979), p. 51.
V. I. Arnold:Usp. Mat. Nauk,18, 13 (1963);Russ. Math. Surv.,18, 9 (1963);Usp. Mat. Nauk.,18, No. 6, 91;Russ. Math. Surv.,18, No. 6, 85 (1963).
V. I. Arnold andA. Avez:Problèmes ergodiques de la mécanique classique, (Paris, 1967).
J. Moser:Proceedings of the International Conference on Functional Analysis and Related Topics (Tokyo, 1969), p. 60;Stable and random motions in dynamical systems (Princeton, 1973).
S. Sternberg:Celestial Mechanics (New York, N. Y., 1968), Part. 2.
E. A. Grebennikov andJu. A. Riabov:New Qualitative Methods in Celestial Mechanics (Moskow, 1971), in Russian.
H. Rüssmann:Nachr. Akad. Wiss. Gott. Math. Phys. Kl., 67 (1970); 1, (1972);Lecture Notes in Physics, No. 38 (Berlin, 1975), p. 598; inSelecta Mathematica, Vol.5, (Heidelberg, Berlin, 1979); inG. Helleman (Editors):Ann. N. Y. Acad. Sci.,357, 90 (1980); see alsoOn the construction of invariant tori of nearly integrable Hamiltonian systems with applications, notes of lectures given at the Institut de Statistique, Laboratoire de Dynamique Stellaire, Université Pierre et Marie Curie, Paris.
R. Barrar:Celes. Mech.,2, 494 (1970).
E. Zehnder:Commun. Pure Appl. Math.,28, 91 (1975);29, 49 (1976); see alsoStability and instability in celestial mechanics, lectures given at Départment de Physique, Laboratoire de Physique Théorique, Ecole Polytechnique Fédérale, Lausanne.
A. I. Neichstadt:J. Appl. Math. Mech.,45, 766 (1981).
G. Gallavotti:Meccanica elementare (Torino, 1980);The Elements of Mechanics (Berlin, 1983); inJ. Frölich (Editor):Progress in Physics (Boston, 1982);L. Chierchia andG. Gallavotti:Nuovo Cimento B,67, 277 (1982).
J. Poeschel:Commun. Pure Appl. Math.,35, 653 (1982);Celest. Mech.,28, 133 (1982).
C. L. Siegel:Ann. Math.,43, 607 (1942).
B. A. Fuchs:Introduction to the Theory of Analytic Functions of Several Complex Variables, Translations of mathematical monographs, Vol.8 (Providence, R. I., 1963).
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Benettin, G., Galgani, L., Giorgilli, A. et al. A proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method. Nuov Cim B 79, 201–223 (1984). https://doi.org/10.1007/BF02748972
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DOI: https://doi.org/10.1007/BF02748972