Geometrical dissipation for dynamical systems

On a Riemannian manifold $(M,g)$ we consider the $k+1$ functions $F_1,\mathrm{.}\mathrm{.}\mathrm{.},F_k,G$ and construct the vector fields that conserve $F_1,\mathrm{.}\mathrm{.}\mathrm{.},F_k$ and dissipate $G$ with a prescribed rate. We study the geometry of these vector fields and prove that they are of gradient type on regular leaves corresponding to $F_1,\mathrm{.}\mathrm{.}\mathrm{.},F_k$. By using these constructions we show that the cubic Morrison dissipation and the Landau-Lifschitz equation can be formulated in a unitary form.