Continuous and discrete damping reduction for systems with quadratic interaction

TL;DR

This paper explores the relationship between Lagrangian and Hamiltonian frameworks in systems with quadratic interactions and damping, introducing a damping reduction method that preserves key features at both continuous and discrete levels, with applications to numerical schemes.

Contribution

It establishes the commutativity of Legendre transform and damping reduction, providing new variational integrators for damped systems that retain important properties of their symplectic counterparts.

Findings

Discrete damping reduction yields numerical schemes that are not symplectic but preserve key features.

Theoretical results are demonstrated with heat bath and transmission line examples.

Simulations show improved performance of variational integrator-based schemes.

Abstract

We study the connection between Lagrangian and Hamiltonian descriptions of closed/open dynamics, for a collection of particles with quadratic interaction (closed system) and a sub-collection of particles with linear damping (open system). We consider both continuous and discrete versions of mechanics. We define the Damping Reduction as the mapping from the equations of motion of the closed system to those of the open one. As variational instruments for the obtention of these equations we use the Hamilton's principle (closed dynamics) and Lagrange-d'Alembert principle (open dynamics). We establish the commutativity of the branches Legendre transform + Damping Reduction and Damping Reduction+Legendre transform, where the Legendre transform is the usual mapping between Lagrangian and Hamiltonian mechanics. At a discrete level, this commutativity provides interesting insight about the resulting integrators. More concretely, Discrete Damping Reduction yields particular numerical schemes for linearly damped systems which are not symplectic anymore, but preserve some of the features of their symplectic counterparts from which they proceed (for instance the semi-implicitness in some cases). The theoretical results are illustrated with the examples of the heat bath and transmission lines. In the latter case some simulations are displayed, showing a better performance of the integrators with variational origin.