Abstract:
Jacobi manifolds provide a unified geometric framework generalising both Poisson and contact manifolds and arise naturally in a wide range of physical systems - from damped oscillators to rigid body dynamics. This talk presents a systematic construction of structure-preserving numerical integrators for Hamiltonian systems on Jacobi manifolds, built upon the theory of homogeneous symplectic groupoids, poissonization, and a Hamilton-Jacobi generating function approach. The resulting Jacobi Hamiltonian Integrators (JHI) achieve high convergence orders while respecting the underlying geometric structure over long time horizons, a property that standard methods such as Runge-Kutta fail to guarantee. We illustrate the theory through several concrete examples, including the damped harmonic oscillator, Lotka-Volterra systems, and rigid body rotation, supported by numerical experiments demonstrating the superiority of the method.