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 $$\mathcal{O}(ρ_0):=\left\{ρ_0\circ Φ\mid Φ\in {\rm DHam}(T^*M)\right\},\quad ρ_0\in L^r(T^*M,\mathbb{R}), \quad r\in[1,\infty),$$ where $ρ_0$ 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 $ρ_0$ are thus contained in $\overline{\mathcal{O}(ρ_0)}$, where the closure is taken with respect to the $L^r$-topology. Our first result is a characterization of $\overline{\mathcal{O}(ρ_0)}$ when the manifold $M$ is the Euclidean space $\mathbb{R}^d$ or the torus $\mathbb{T}^d$ of arbitrary dimension: $\overline{\mathcal{O}(ρ_0)}$ is the set of all the densities whose sub- and super-level sets have the same measure as those of $ρ_0$. 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.