Let us have a Riemmanian manifold $(M,g)$. Is there some “natural“/canonical way to lift the metric to the cotangent bundle $T^* M$, i.e. define a metric on $T^*M$?
I ask because I have read some articles on orthogonal separation of variables (on configuration space) heavily using Killing tensors and other notions from Riemannian geometry and I am thinking whether it can be extended to separation on the phase space of Hamiltonian mechanics.