Jacobi Metric and Morse Theory of Dynamical Systems
Mathematical Physics
2007-05-23 v1 High Energy Physics - Theory
math.MP
Abstract
The generalization of the Maupertuis principle to second-order Variational Calculus is performed. The stability of the solutions of a natural dynamical system is thus analyzed via the extension of the Theorem of Jacobi. It is shown that the Morse Theory of the trajectories in the dynamical system is identical to the Morse Theory of geodesics in the Jacobi metric, even though the second-variation functionals around the action and the Jacobi length do not coincide. As a representative example, we apply this result to the study of the separatrix solutions of the Garnier System.
Cite
@article{arxiv.math-ph/0212017,
title = {Jacobi Metric and Morse Theory of Dynamical Systems},
author = {A. Alonso Izquierdo and M. A. Gonzalez Leon and J. Mateos Guilarte and M. de la Torre Mayado},
journal= {arXiv preprint arXiv:math-ph/0212017},
year = {2007}
}
Comments
11 pages, 3 figures. Talk delivered at the XI Fall Workshop on Geometry and Physics held at Oviedo, Spain, 2002