Consider a compact, not simply connected manifold which is the configuration space of a mechanical system (e.g. a torus, the configuration space of a linear double pendulum).
Some well-known theorems say that on such manifold there exists at least one (closed) geodesic for every closed curves homotopy class (provided the total energy of the system is high enough).
What puzzles me is that (in general) such geodesics could in fact intersect transversally. Either with themselves, either with one another.
Does this mean that also in phase space such intersection could occur?