I have a Lie group action $\varphi$ of $G$ on $TM$ (tangent bundle of $M$) and I want to define an associated action on $T^*M$ (cotangent bundle of $M$), where $M$ is a smooth manifold. I know that $TM$ and $T^*M$ are diffeomorphic (but I don't think I have seen before an explicit expression for this mapping). Thus what I think to do is define the action $$\psi:G\times T^*M\rightarrow T^*M $$ by means of this diffeomorphism $f:T^*M\rightarrow TM$ as $\psi(g,m) := \varphi(g,f(m)) = f^*\varphi_g(m)$.
I think this is a reasonable approach, however before going on I want to be sure this is correct. It would be even helpful if someone tells me the explicit expression of this $f$, or how to construct it.