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.