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 a paper to appear soon [19] 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 [49], where the discrete flow is defined on Q).…”

Lagrangian reduction of forced discrete mechanical systems

“…Recall that a forced Lagrangian mechanical system consists of a triple (Q, L, f), where Q is a smooth manifold, the configuration space, L : TQ −→ R is a smooth map, the Lagrangian, and f is a horizontal 6 1-form on TQ, the force 7 .…”

“…Given a forced mechanical system 6 A differential form on the total space of a fiber bundle ϕ : Q −→ M is called horizontal if it vanishes when any of its arguments is a vertical vector, i.e. an element of ker Tϕ.…”

“…Forced discrete mechanical systems have been known and studied for a number of years (see Part Three of [15]). Recently, the interest in these systems has been growing (see, for instance, [6,7,16,17]).…”

“…Other work in the study of symmetries of forced discrete mechanical system includes [6,15], where Noether-type theorems are studied, but they do not go as far as to present a notion of reduced system.…”