A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equa...

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Vydáno v:	Journal of nonlinear science Ročník 27; číslo 2; s. 495 - 530
Hlavní autor:	Villanueva, Jordi
Médium:	Journal Article Publikace
Jazyk:	angličtina
Vydáno:	New York Springer US 01.04.2017 Springer Nature B.V
Témata:	37 Dynamical systems and ergodic theory 37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems 70 Mechanics of particles and systems 70K Nonlinear dynamics Analysis Classical Mechanics Classificació AMS Economic Theory/Quantitative Economics/Mathematical Methods Errors Functional equations Hamiltonian functions Hamiltonian systems Invariance Invariants KAM theory Kolmogorov-Arnold-Moser theory Lagrangian tori Matemàtiques i estadística Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Parameterization Parameterization methods Perturbation methods Quasi Newton methods Sistemes hamiltonians Theoretical Toruses Àrees temàtiques de la UPC Hamiltonian systems 37J40 Lagrangian tori 70K43 Parameterization methods KAM theory
ISSN:	0938-8974, 1432-1467
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Popis
Shrnutí:	We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorov’s non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of O ( ε 1 / 2 ) , where ε is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.
Bibliografie:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0938-8974 1432-1467
DOI:	10.1007/s00332-016-9342-5