Discrete Hamilton-Jacobi theory for systems with external forces

Abstract

This paper is devoted to discrete mechanical systems subject to external forces. We introduce a discrete version of systems with Rayleigh-type forces, obtain the equations of motion and characterize the equivalence for these systems. Additionally, we obtain a Noether's theorem and other theorem characterizing the Lie subalgebra of symmetries of a forced discrete Lagrangian system. Moreover, we develop a Hamilton-Jacobi theory for forced discrete Hamiltonian systems. These results are useful for the construction of so-called variational integrators, which, as we illustrate with some examples, are remarkably superior to the usual numerical integrators such as the Runge-Kutta method.

Geometric Hamilton–Jacobi theory for systems with external forces

“…In another paper, 60 we develop a Hamilton-Jacobi theory for forced discrete Hamiltonian systems. Our approach is based on the construction of a discrete flow on Q × Q (unlike the case without external forces, 61 where the discrete flow is defined on Q).…”