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.
$$ \mathcal{L}_\mathbf{v}(g) $$
I would appreciate your help!