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 $f_\#^*\alpha=\alpha$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 140 / 3 / a / Solution

{title2=$f_\#$}

For a diffeomorphism $f:X\to X$, its cotangent lift is
$$
f_\#(x,\xi)=\bigl(f(x),(df_x^{-1})^*\xi\bigr).
$$
It preserves the canonical one-form and therefore the canonical symplectic form. Cotangent lift is functorial under composition.


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