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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Bibliographic Details
Published in:Journal of nonlinear science Vol. 27; no. 2; pp. 495 - 530
Main Author: Villanueva, Jordi
Format: Journal Article Publication
Language:English
Published: New York Springer US 01.04.2017
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-016-9342-5