Morse theory for conditionally-periodic solutions to Hamiltonian systems
type: Item
instance of: scholarly article
title: Morse theory for conditionally-periodic solutions to Hamiltonian systems
zbMATH Open document ID: 0859.58024
DOI: 10.1007/BF02104825
author: P. I. Plotnikov
published in: Siberian Mathematical Journal
publication date: 1997-04-07T00:00:00Z
review text: The author studies perturbations of integrable Hamiltonian systems; i.e., consider the systems of differential equations \[ \dot x=\nabla_yH,\quad \dot y=-\nabla_xH\tag{1} \] with analytic Hamiltonians of the form \[ H(x,y)=H_0(y)+\varepsilon H_1(x,y),\quad (x,y)\in T^n\times D. \] Here \(x\) and \(y\) are the variables ``angle-action''. The main purpose of this article is to develop a local Morse theory for the existence problem of invariant tori of (1), which is reduced to a problem for finding critical points of a smooth function on a finite-dimensional manifold. The present method differs from the traditional method of accelerated convergence in the KAM theory, and leans on the Nash-Moser implicit function theorem, especially on the version by E. Zehnder. The author also demonstrates some applications of theorems obtained in the paper to proving existence of conditionally-periodic solutions of (1).
reviewed by: Ding Tongren
Mathematics Subject Classification ID: 37C55
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 917077
zbMATH Keywords: perturbations
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: local Morse theory
zbMATH Keywords: invariant tori
zbMATH Keywords: KAM theory
MaRDI profile type: Publication
cites work: Q5829223
cites work: Q5570390
cites work: Q3961135
cites work: Q3031274
cites work: Q3861996
cites work: Continuation to the minimal number of critical points in gradient flows
cites work: Generalized implicit function theorems with applications to some small divisor problems, I
full work available at URL: https://doi.org/10.1007/bf02104825
OpenAlex ID: W2088562753
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description: scientific article; zbMATH DE number 917077