Jacobi Hamiltonian Integrators

We develop a method of constructing structure-preserving integrators for Hamiltonian systems in Jacobi manifolds. Hamiltonian mechanics, rooted in symplectic and Poisson geometry, has long provided a foundation for modelling conservative systems in classical physics. Jacobi manifolds, generalizing both contact and Poisson manifolds, extend this theory and are suitable for incorporating time-dependent, dissipative and thermodynamic phenomena. Building on recent advances in geometric integrators - specifically Poisson Hamiltonian Integrators (PHI), which preserve key features of Poisson systems - we propose a construction of Jacobi Hamiltonian Integrators. Our approach explores the correspondence between Jacobi and homogeneous Poisson manifolds, with the aim of extending the PHI techniques while ensuring preservation of the homogeneity structure. This work develops the theoretical tools required for this generalization and outlines a numerical integration technique compatible with Jacobi dynamics. By focusing on the homogeneous Poisson perspective rather than on direct contact realizations, we provide a clear pathway for structure-preserving integration of time-dependent and dissipative systems within the Jacobi framework.