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Journal of Dynamical Systems and Geometric Theories

(Since 2002)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0057·ISSN (Print): 1726-037X

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Half-Yearly Journal: Publishes research on dynamical systems and geometry, including Random Dynamical Systems, hyperbolic behaviour, complex geometry and Ergodic theory.

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Journal of Dynamical Systems and Geometric Theories
Open Access Article

Hamiltonian dynamical systems and geometry of surfaces in 3-D

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pp. 163–176Vol. 15Issue 2July 2017DOI: 10.1080/1726037X.2017.1390847XML
Received:
28 Feb 2016
Accepted:
23 Sep 2017
Published Online:
06 Dec 2017
Article type:
Article
Language:
EN
Article no.:
1726037X.2017.1390847
Pages:
163–176

Abstract

Hamiltonian vector field, Poisson vector field and the gradient of Hamiltonian function defines Darboux frame along an integral curve of a Hamiltonian dynamical system on a surface whose normal vector field corresponds to the Poisson structure for a given Hamiltonian system. We show that the existence of compatible Poisson structures determined by the normal legs of the Darboux frame is resolved to the characteristic equation for the Weingarten map. We also show that a Hamiltonian dynamical system in three dimensions has bi-Hamiltonian representation determined by the normal legs of Frenet-Serret triad if and only if an integral curve of Hamiltonian vector field is both a geodesic and a line of curvature.

Keywords

Hamiltonian dynamical systemsDarboux framegeodesic curvatureWeingarten mapcompatible Poisson structuresbi-Hamiltonian representation

Subject Classifications

53A0453A0537K1037K25

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