# Producing term with Lie derivative from variational principle

Is it possible to obtain variational principle (Lagrangian) such that the equations of motions contain Lie derivative? For example, if $$g$$ is a standard metric tensor in Euclidean space $$E^3$$ and we have some vector field $$\mathbf{v}$$, and Lie derivative is denoted by $$\mathcal{L}$$, then the variational principle produces term that is of the following form.

$$\mathcal{L}_\mathbf{v}(g)$$