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Chaotic trajectories for natural systems on a torus

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


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