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September  2003, 9(5): 1343-1357. doi: 10.3934/dcds.2003.9.1343

Chaotic trajectories for natural systems on a torus

M. L. Bertotti 1, and  Sergey V. Bolotin 2,

1. 

Department of Mathematics and Applications, University of Palermo, Palermo, Italy
2. 

Department of Mathematics, University of Wisconsin, Madison, United States

Received  July 2002 Revised  December 2002 Published  June 2003

We consider a natural Lagrangian system on a torus and give sufficient conditions for the existence of chaotic trajectories for energy values slightly below the maximum of the potential energy. It turns out that chaotic trajectories always exist except when the system is "variationally separable", i.e. minimizers of the action functional behave like in a separable system. This gives some more support for an old conjecture that only separable natural Lagrangian systems on a torus are integrable.
Keywords: Natural Lagrangian systems, shadowing trajectory., Maupertuis principle, homoclinic orbit, chaotic trajectory, Jacobi metric.
Mathematics Subject Classification: 37J45, 37C29, 34C37, 70H30, 58E1.
Citation: M. L. Bertotti, Sergey V. Bolotin. Chaotic trajectories for natural systems on a torus. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1343-1357. doi: 10.3934/dcds.2003.9.1343
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