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On normal forms in Hamiltonian dynamics, a new approach to some convergence questions

Published online by Cambridge University Press:  19 September 2008

David Delatte
David Delatte
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-0116, USA
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Abstract

We consider the convergence of solutions to various conjugacy problems occurring in Hamiltonian dynamics. These give rise to nonlinear functional equations which we solve by using a norm well adapted to the problem. The approach provides an alternative to the classical majorization method.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 1 , February 1995 , pp. 49 - 66
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[A]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds with negative curvature. Tr. Mat. Inst. Steklov. 90 (1967) (Russian); Proc. Steklov Inst. Math. 90 (1967).Google Scholar
[Ar]Arnold, V. I.. Proof of A. N. Kolmogorov's theorem on the presentation of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18: 6 (1963), 85–192.Google Scholar
[D]DeLatte, D. A.. Nonstationary normal forms and cocycle invariants. Random and Comp. Dyn. 1: 2 (1992–1993), 229–259.Google Scholar
[H]Hoffman, K.. Banach Spaces of Analytic Functions. Prentice Hall: Englewood Cliffs, NJ, 1962.Google Scholar
[K]Katok, A. B.. Dynamical systems with hyperbolic structure; three papers on dynamical systems. Amer. Math. Soc. Transl. 116 (1981), 125–211.Google Scholar
[Ko]Kolmogorov, A. N.. Dokl. Akad. Nauk 98 (1954), 527–530 (Russian).Google Scholar
[Ma]Mañé, R.. Ergodic Theory and Differentiable Dynamics. Springer: Berlin, 1987.CrossRefGoogle Scholar
[Me]Meyer, K. R.. The implicit function theorem and analytic differential equations. Springer Lecture Notes in Mathematics, 468 (1974), 191–208.CrossRefGoogle Scholar
[M1]Moser, J. K.. The analytic invariants of an area-preserving mapping near a hyperbolic fixed point. Comm. Pure Appl. Math. 9 (1956), 673–692.CrossRefGoogle Scholar
[M2]Moser, J. K.. On the generalization of a theorem of Liapunoff. Comm. Pure Appl. Math. 11 (1958), 257–275.CrossRefGoogle Scholar
[M3]Moser, J. K.. On invariant curves of area-preserving mappings of the annulus. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl. IIa: 1 (1962), 1–20.Google Scholar
[M4]Moser, J. K.. A rapidly converging iteration method and nonlinear partial differential equations I, II. Ann. Scuola. Norm. Sup. Pisa 320 (1966), 265–315. 499–535.Google Scholar
[MW]Moser, J. K. and Webster, S.. Normal forms for real surfaces in ℂ2 near complex tangents and hyperbolic surface transformations. Acta. Math. 150 (1983), 255–296.CrossRefGoogle Scholar
[N]Nirenberg, L.. On elliptic partial differential equations. Ann. Sc. Norm. Sup. Pisa 13 (1959), 1–48.Google Scholar
[S]Siegel, C. L.. Vereinfachter Beweis eines Satzes von J. Moser. Comm. Pure Appl. Math. 10 (1957), 305–309.CrossRefGoogle Scholar
[Sc]Schauder, J.. Das Anfangswertproblem einer quasi linearen hyperbolischen Differentialgleichung zweiter Ordnung in beliebiger Anzahl von unabhängigen Veränderlichen. Fund. Math. 24 (1935), 213–246.CrossRefGoogle Scholar
[S-M]Siegel, C. L. and Moser, J. K.. Lectures on Celestial Mechanics. Springer: New York, 1971.CrossRefGoogle Scholar