Jacobi Metric and Morse Theory of Dynamical Systems

Details Jacobi Metric and Morse Theory of Dynamical Systems Jacobi Metric and Morse Theory of Dynamical Systems

2002-12-04

arxiv.org/abs/math-ph/0212017v1

Publ. R. Soc. Mat. Esp. 6 (2004) 81--91

Physics

Mathematical Physics

11 pages, 3 figures. Talk delivered at the XI Fall Workshop on Geometry and Physics held at Oviedo, Spain, 2002

Scientific paper

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.

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