Stability of equilibrium points for a Hamiltonian systems with one degree of freedom in one degenerate case
type: Item
instance of: scholarly article
title: Stability of equilibrium points for a Hamiltonian systems with one degree of freedom in one degenerate case
zbMATH Open document ID: 1403.37066
DOI: 10.1134/S1560354717070097
author: Rodrigo Gutierrez
author: Claudio Vidal
published in: Regular and Chaotic Dynamics
publication date: 2018-10-16T00:00:00Z
review text: Hamiltonian systems have a wide application in theoretical physics and celestial mechanics. Some important topics in Hamiltonian theory are existence of quasi-periodic, periodic and stationary solutions, and stability of equilibrium points. The main theoretical results are combined in the Kolmogorov-Arnold-Moser theory. The authors consider one degree of freedom Hamiltonian systems such that the origin is an equilibrium solution and the Hamiltonian function \(H=H(q,p,t)\) is a real analytic function of \((q,p)\), \(2\pi\)-periodic in \(t\), in a neighborhood of the equilibrium point. Additionally, it is assumed that the Taylor series of \(H\) has the form \(H=H_2+H_3+\dots+H_j+\dots\), where \(H_j\) is an homogeneous polynomial of degree \(j\) in \((q,p)\). The so-called degenerate case occurs when the point \((0,0)\) is the equilibrium point of the above Hamiltonian system. The cases \(j=4,5,6\) are considered in detail. Some new results connected with the stability problem are found and discussed.
reviewed by: Yulianna Perepelkina
Mathematics Subject Classification ID: 37J25
Mathematics Subject Classification ID: 70H14
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 6955785
zbMATH Keywords: stability problem
zbMATH Keywords: perturbations
zbMATH Keywords: normal forms
zbMATH Keywords: small divisors
zbMATH Keywords: KAM theory
zbMATH Keywords: Arnol'd diffusion
MaRDI profile type: Publication
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full work available at URL: s1560354717070097
OpenAlex ID: W2793639673
Wikidata QID: Q130203257
description: scientific article; zbMATH DE number 6955785