Invariant tori through direct solution of the Hamilton-Jacobi equation
type: Item
instance of: scholarly article
title: Invariant tori through direct solution of the Hamilton-Jacobi equation
zbMATH Open document ID: 0612.58017
DOI: 10.1016/0167-2789(87)90213-2
published in: Physica D
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://www.osti.gov/biblio/1447885
review text: We explore a method to compute invariant tori in phase space for classical non-integrable Hamiltonian systems. The procedure is to solve the Hamilton-Jacobi equation stated as a system of equations for Fourier coefficients of the generating function. The system is truncated to a finite number of Fourier modes and solved numerically by Newton's method. The resulting canonical transformation serves to reduce greatly the non- integrable part of the Hamiltonian. Successive transformations computed on progressively larger mode sets would lead to exact invariant tori, according to the argument of Kolmogorov, Arnol'd, and Moser (KAM). The procedure accelerates the original KAM algorithm since each truncated Hamilton-Jacobi equation is solved accurately, rather than in lowest order. In examples studied to date the convergence properties of the method are excellent. One can include enough modes at the first stage to get accurate results with only one canonical tranformation. The method is effective even on the borders of chaotic regions and on the separatrices of isolated broad resonances. We propose a criterion for the transition to chaos and verify its utility in an example with \(1{1/2}\) degrees of freedom. We anticipate that the criterion will be useful as well in systems of higher dimension.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70H20
zbMATH DE Number: 3990437
zbMATH Keywords: KAM-theory
zbMATH Keywords: non-integrable Hamiltonian systems
zbMATH Keywords: Hamilton-Jacobi equation
author: Robert Lee Warnock
author: Ronald D. Ruth
MaRDI profile type: Publication
OpenAlex ID: W1972416632
cites work: Q4167627
cites work: Q3657476
cites work: Regular and stochastic motion
cites work: Maxwell equations in nonuniformly moving media
cites work: Q5638711
cites work: Q5332416
cites work: Integrability of hamiltonian systems on cantor sets
cites work: Q4026085
cites work: Q5578455
cites work: Q5615660
cites work: FFT algorithms for vector computers
cites work: On Solving Nonlinear Equations with Simple Singularities or Nearly Singular Solutions
cites work: Widely Convergent Method for Finding Multiple Solutions of Simultaneous Nonlinear Equations
cites work: An extension of the Newton-Kantorovic method for solving nonlinear equations with an application to elasticity
cites work: Galerkin's procedure for nonlinear periodic systems
cites work: Q4114704
cites work: Hamiltonian maps in the complex plane
Recommended article: Q5474665
Recommended article: Q3474921
Recommended article: Invariant tori for commuting Hamiltonian PDEs
Recommended article: Invariant tori of intermediate dimensions in Hamiltonian systems
Recommended article: Q4240827
Recommended article: Invariant hyperbolic tori for Hamiltonian systems with degeneracy
Recommended article: Q4727896
Recommended article: Invariant tori in non-degenerate nearly integrable Hamiltonian systems
Recommended article: Q3976420
Recommended article: Existence of higher dimensional invariant tori for Hamiltonian systems
description: scientific article; zbMATH DE number 3990437