Liouville vector field (${\iota }_Z\omega =\lambda ,{\mathcal{L}}_Z\omega =\omega$)

For an exact symplectic manifold with $\omega =d\lambda$, the Liouville vector field is defined by ${\iota }_Z\omega =\lambda$. Cartan's magic formula gives ${\mathcal{L}}_Z\omega =\omega$. On a cotangent bundle using $\omega =-d\alpha$, take $\lambda =-\alpha$; then $Z={\sum }_i{p}_i{\partial }_{p_i}$ generates the cotangent fiber-dilation flow.

The vertical Liouville vector field on a cotangent bundle has complete flow $D_s(x,p)=(x,e^{s}p)$. Its pullbacks scale both the Liouville one-form and the canonical symplectic form by $e^s$. As $s\to -\infty$, each orbit tends to its base point on the zero section. This contraction proves rigidity of cotangent Liouville-form preservation.