= Solution
The characteristic vector field $b=(\nabla_vH,-\nabla_xH)$ is a <divergence-free vector field>:
$$
\nabla_z\cdot b=\sum_{i=1}^n(H_{x_iv_i}-H_{v_ix_i})=0.
$$
The <Jacobian determinant> $J_t=\det D\Phi_t$ obeys $\dot J_t=(\nabla\cdot b)(\Phi_t)J_t=0$, with $J_0=1$. Thus the flow preserves phase-space volume, the <Liouville theorem in Hamiltonian mechanics>. Changing variables $(x,v)=\Phi_t(\xi,\eta)$ and using (i) gives
$$
\boxed{\int_{\mathbb R^{2n}}F(f(x,v,t))\,dx\,dv=\int_{\mathbb R^{2n}}F(f_I(\xi,\eta))\,d\xi\,d\eta.}
$$
This applies on intervals where the flow is a diffeomorphism of the phase space and the integrals are defined; for finite conservation statements require integrability. For instance compactly supported initial data and $F(0)=0$ give the usual setting. An arbitrary smooth $F$ need not make these integrals finite on infinite-volume space, and merely smooth $H$ does not exclude escape of characteristics to infinity. Equivalently $F(f)_t+\nabla_z\cdot[bF(f)]=0$, with no flux through infinity, proves the same <Casimir invariant of a transport equation> conservation.
Back to article page
