On the density of nonintegrable Hamiltonian systems close to being billiard
type: Item
instance of: scholarly article
title: On the density of nonintegrable Hamiltonian systems close to being billiard
zbMATH Open document ID: 0991.37037
author: S. I. Pidkujko
published in: Matematychni Studiï
publication date: 2002-07-21T00:00:00Z
review text: The author considers Hamiltonian systems with the Hamiltonian \(H=\frac 12(p_1^2+p_2^2)+V(f(x_1,x_2))\), where \(f(x_1,x_2)\) is a polynomial and the equation \(f(x_1,x_2)=0\) determines a closed curve in \(\mathbb{R}^2\) which bounds a certain region. If \(V\) is a meromorphic function in a neighborhood of zero that has a pole at zero, then we have a one-parameter family of Hamiltonian systems that is close to the billiard system. It is proved that the nonintegrable Hamiltonian systems from this class form an everywhere dense subset. It is an analogue of the famous Siegel theorem [see \textit{C. L. Siegel}, Math. Ann. 128, 144-170 (1954; Zbl 0057.32002)].
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 37K10
zbMATH DE Number: 1769802
zbMATH Keywords: nonintegrable Hamiltonian system
zbMATH Keywords: billiard system
zbMATH Keywords: density
reviewed by: M. P. Moklyachuk
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1769802