Cotangent lift of a diffeomorphism ($f_{\#}$)

For a diffeomorphism $f:X\to X$, its cotangent lift isIt preserves the canonical one-form and therefore the canonical symplectic form. Cotangent lift is functorial under composition.

If $x=\chi (x^{\prime })$ is an angular diffeomorphism and $\eta \cdot dx$ is a closed differential one-form, the map $y=\eta (\chi (x^{\prime }))+D\chi (x^{\prime }{)}^{-T}{y}^{\prime }$ is a symplectomorphism. It combines a cotangent lift of a diffeomorphism with translation by a closed differential one-form. In particular $\eta =du+a$ is allowed on a torus even when the constant one-form $a\cdot dx$ is not exact.

A diffeomorphism of the entire cotangent bundle preserving the Liouville one-form is the cotangent lift of a diffeomorphism of the base. It preserves the zero section and the Liouville vector field, hence commutes with the cotangent fiber-dilation flow. The limit as the fiber contracts forces it to cover its restriction to the zero section. Evaluation of the preserved one-form then forces the inverse-transpose derivative action on each covector.