Energy-preserving Variational Integrators for Forced Lagrangian Systems
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.
Cite
@article{arxiv.1801.04996,
title = {Energy-preserving Variational Integrators for Forced Lagrangian Systems},
author = {Harsh Sharma and Mayuresh Patil and Craig Woolsey},
journal= {arXiv preprint arXiv:1801.04996},
year = {2018}
}