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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 710274, 26 pages
Research Article

Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

1Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK
2Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
3School of Automation, Beijing Institute of Technology, Beijing 100081, China

Received 3 December 2010; Revised 19 March 2011; Accepted 12 April 2011

Academic Editor: Heinrich Begehr

Copyright © 2011 Linyu Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.