Closed characteristics on non-compact hypersurfaces in \(\mathbb {R}^{2n}\)
type: Item
instance of: scholarly article
title: Closed characteristics on non-compact hypersurfaces in \(\mathbb {R}^{2n}\)
zbMATH Open document ID: 1178.53084
author: Federica Pasquotto
author: Jan Bouwe Van den Berg
author: Robert Vandervorst
published in: Mathematische Annalen
publication date: 2009-02-09T00:00:00Z
review text: The authors consider mechanical Hamiltonians \(H(p,q)=1/2 |p|^2 +V(q)\) on the standard symplectic vector space \(({\mathbb R}^{2n}, \omega).\) Assuming that \(0\) is a regular value of \(H\) and the level hypersurface \(M=H^{-1}(0)\) asymptotically regular they prove that \(M\) contains at least one closed characteristic of the Hamiltonian vectorfield associated to \(H\) provided that the homology of \(M\) is nontrivial on dimensions greater or equal to \(n\). Roughly speaking, the asymptotic regularity requires that the growth of the Hessian of the potential function \(V\) is dominated by that of the gradient of \(V\) at infinity. This condition is needed in order to be able to deal with non-compact hypersurfaces. The proof of the above result, as many others of this type, relies on a linking principle. However due to the lack of compactness of \(M\) the technical details are considerably more involved than in the compact case.
reviewed by: Jacobo Pejsachowicz
Mathematics Subject Classification ID: 53D05
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53D10
zbMATH DE Number: 5503660
MaRDI profile type: Publication
full work available at URL: s00208-008-0271-y
OpenAlex ID: W1982652822
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DOI: 10.1007/S00208-008-0271-Y
description: scientific article; zbMATH DE number 5503660