Discrete Hamilton-Jacobi theory for systems with external forces

TL;DR

This paper develops a discrete Hamilton-Jacobi theory for forced systems, extending variational integrator construction and symmetry analysis, demonstrating improved numerical performance over traditional methods.

Contribution

It introduces a discrete Hamilton-Jacobi framework for forced systems, including new equations, symmetry characterizations, and applications to variational integrators.

Findings

Discrete Hamilton-Jacobi theory for forced systems

Symmetry and Noether's theorem for discrete forced systems

Variational integrators outperform Runge-Kutta methods

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.