Local Universal Splitting Integrators for Contact Hamiltonian Systems

Contact Hamiltonian systems extend symplectic Hamiltonian mechanics to dissipative settings while retaining geometric structure. We develop a structure-preserving splitting framework for contact Hamiltonian systems on $J^1({\mathbb{R}}^n)$ based on two tractable classes of exact-contact subflows: strict contactomorphisms and prolonged diffeomorphisms. Our main theoretical result is that the Lie algebra generated by the corresponding strict and prolonged Hamiltonians contains all polynomial-in-$p$ Hamiltonians and is therefore dense, in the $C^r$ topology on compact sets, in the Lie algebra of smooth contact Hamiltonians. This yields a local universality result and contact splitting integrators built from exact strict and prolonged subflows. We then show how these subflows can be realized numerically by lifting symplectic integrators on $T^{\ast }{\mathbb{R}}^n$ and ODE integrators on ${\mathbb{R}}^n\times \mathbb{R}$. Finally, we illustrate the framework on a sequence of low-dimensional examples.