Orbits and attainable Hamiltonian diffeomorphisms of mechanical Liouville equations

We study the approximate controllability problem for Liouville transport equations along a mechanical Hamiltonian vector field. Such PDEs evolve inside the orbit $$\mathcal{O}(\rho_0):=\left\{\rho_0\circ \Phi\mid \Phi\in {\rm DHam}(T^*M)\right\},\quad \rho_0\in L^r(T^*M,\mathbb{R}), \quad r\in[1,\infty),$$ where $\rho_0$ is the initial density and ${\rm DHam}(T^*M)$ is the group of Hamiltonian diffeomorphisms of the cotangent bundle manifold $T^*M$. The approximately reachable densities from $\rho_0$ are thus contained in $\overline{\mathcal{O}(\rho_0)}$, where the closure is taken with respect to the $L^r$-topology. Our first result is a characterization of $\overline{\mathcal{O}(\rho_0)}$ when the manifold $M$ is the Euclidean space $\mathbb{R}^d$ or the torus $\mathbb{T}^d$ of arbitrary dimension: $\overline{\mathcal{O}(\rho_0)}$ is the set of all the densities whose sub- and super-level sets have the same measure as those of $\rho_0$. This result is an approximate version, in the case of ${\rm DHam}(T^*M)$, of a theorem by J. Moser (Trans. Am. Math. Soc. 120: 286-294, 1965) on the group of diffeomorphisms. We then present two examples of systems, respectively on $M=\mathbb{R}^d$ and $\mathbb{T}^d$, where the small-time approximately attainable diffeomorphisms coincide with ${\rm DHam}(T^*M)$, respectively at the level of the group and at the level of the densities. The proofs are based on the construction of Hamiltonian diffeomorphisms that approximate suitable permutations of finite grids, and Poisson bracket techniques.