Solution
= Solution
Define the <cotangent lift of a diffeomorphism>
$$
f_\#(x,\xi)=\left(f(x),(df_x^{-1})^*\xi\right).
$$
If $\alpha$ is the canonical one-form and $W\in T_{(x,\xi)}T^*X$, then
$$
(f_\#^*\alpha)_{(x,\xi)}(W)
=((df_x^{-1})^*\xi)(df_x(d\pi W))
=\xi(d\pi W)=\alpha_{(x,\xi)}(W).
$$
Hence $f_\#^*\alpha=\alpha$.