Corresponding flow on cotangent bundle

Question

I have an exercise where it states:

Let $Y:Q \rightarrow TQ$ be a complete vector field with flow $\varphi_t$. Let $X : T^{*}Q → T(T^{*}Q)$ be the vector field generating the corresponding flow $\Phi_t$ on $T^{*}Q$.

Now I don't what is meant by "corresponding flow", how is this flow defined?

Answer 1

The question is talking about the cotangent lift of the flow on $Q$: for any diffeomorphism $f:Q\to Q$, there's a corresponding cotangent lift $T^*f:T^*Q\to T^*Q$, defined by $$ T^*f(\alpha) := \alpha\circ T(f^{-1}) $$ (so $T^*f$ maps $T^*_qQ$ to $T_{f(q)}^*Q$). Apply this to the flow $\varphi_t$ generated by $Y$, i.e. $\Phi_t = T^*\varphi_t$.