Chaotic trajectories for natural systems on a torus

M. L. Bertotti; Sergey V. Bolotin

doi:10.3934/dcds.2003.9.1343

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2003, Volume 9, Issue 5: 1343-1357. Doi: 10.3934/dcds.2003.9.1343

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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
2.
Department of Mathematics, University of Wisconsin, Madison

Received: June 30, 2002

Revised: November 30, 2002

Published: August 2003

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Abstract

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.

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Mathematics Subject Classification: 37J45, 37C29, 34C37, 70H30, 58E10.

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