Evolution of functions

Question

Let $M$ be a manifold and $X$ a vector field. The Lie brackets of functions are defined by:

$$[f,g]_X =f. Xg+ Xf.g$$

I define an evolution of functions by a Hamiltonian $H$:

$$\frac{\partial f}{\partial t}= [f,H]_X$$

Does this evolution correspond to a physical system?