Global characteristic flow for a Hamiltonian with bounded Hessian ($\Vert D_z^{2}H(t,z)\Vert \le L_T$)

If $H$ is $C^2$ and its spatial Hessian matrix is uniformly bounded on every finite time interval, its Hamiltonian vector field is globally Lipschitz continuous in phase space with at most linear growth. The Picard-Lindelöf theorem and Gronwall inequality give a unique Hamiltonian flow for all finite forward and backward times. Bounds on ${\nabla }_{z}H(t,0)$ follow from time continuity on compact intervals.