I feel like there should either be a straightforward example of such a thing, or a homological reason why it can't exist. The reason why I ask is that if the Hessian matrix of a Lagrangian is non-singular then its dynamics are well defined, and (locally) you can pass to a Hamiltonian via the Legendre transformation, but it seems possible that two velocities (tangent vectors) could have the same momentum (cotangent vector, associated via the Legendre transform). In this case you'd have to formulate the Hamiltonian version on a cover of the cotangent bundle, instead of the cotangent bundle itself. Does this situation occur?

propergeneric smooth map $\Bbb R^n\to\Bbb R^n$ without critical points must be one-to-one (injective) and onto (surjective). – Sergey Melikhov May 17 '12 at 8:33