Energy-preserving Variational Integrators for Forced Lagrangian Systems

TL;DR

This paper develops energy-preserving variational integrators for forced, time-dependent mechanical systems, using extended Lagrangian mechanics and adaptive time-stepping to improve numerical performance and energy conservation.

Contribution

It introduces a novel framework for adaptive energy-preserving variational integrators applicable to forced, time-dependent systems, extending discrete mechanics with new theoretical and numerical insights.

Findings

Adaptive time-stepping improves energy preservation in conservative systems.

Implicit equations become more ill-conditioned with larger time steps.

Energy behavior in dissipative systems can be unexpected with adaptive steps.

Abstract

The goal of this paper is to develop energy-preserving variational integrators for time-dependent mechanical systems with forcing. We first present the Lagrange-d'Alembert principle in the extended Lagrangian mechanics framework and derive the extended forced Euler-Lagrange equations in continuous-time. We then obtain the extended forced discrete Euler-Lagrange equations using the extended discrete mechanics framework and derive adaptive time step variational integrators for time-dependent Lagrangian systems with forcing. We consider two numerical examples to study the numerical performance of energy-preserving variational integrators. First, we consider the example of a nonlinear conservative system to illustrate the advantages of using adaptive time-stepping in variational integrators. We show a trade-off between energy-preserving performance and accurate discrete trajectories while choosing an initial time step. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step increases through a condition number analysis. As a second example, we numerically simulate a damped harmonic oscillator using the adaptive time step variational integrator framework. The adaptive time step increases monotonically for the dissipative system leading to unexpected energy behavior.