A Newtonian system mathematically always a trivial bundle?

[pellman]

The phase space of a Newtonian system is a cotangent bundle, where the base space is the manifold M of the configuration space (the positions) and the typical fiber is the cotangent space T*M (the momenta). Is it always the case that this cotangent bundle is the trivial bundle M x TM?

 

[homeomorphic]

No. The cotangent bundle is homeomorphic to the tangent bundle and tangent bundles are not always trivial. M cross TM is the manifold cross the tangent bundle, which is not what you want to say there.