Geometric meaning of Liouville vector field.

Question

A vector field $v$ on a symplectic manifold $(X,\omega)$ is a Liouville vector field if $\mathcal{L}_v\omega=\omega.$

Can anyone explain a geometric meaning of it?

Answer 1

I'm not sure if you are asking this, but if $\phi_t$ is the flow of the vector field $v$ then $\phi_t^*\omega = e^t \omega$. You can think of this as saying that the vector field is the infinitesimal generator of an $\mathbb{R}$ action on the manifold that acts on the symplectic form by scaling it.

(A simple example is of $\mathbb{R}^{2n}$. If you take the Liouville vector field $v = \sum_{i=1}^n \frac{1}{2} \left ( x_i \partial_{x_i} + y_i \partial_{y_i} \right ) $, your vector field is the infinitesimal generator of the action given by actual scaling of $\mathbb{R}^{2n}$.)

(Edit: thanks to Filip Brocic for pointing out the error in the original version of this answer. My formula was wrong, though the geometric interpretation matches the correct answer that is now present.)