Energy-preserving variational integrators for forced Lagrangian systems

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 three 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. In addition, we demonstrate how the implicit equations become more ill-conditioned as the adaptive time step decreases through a condition number analysis. As a second example, we numerically simulate the time-dependent example of a forced harmonic oscillator to demonstrate the superior energy performance of energy-preserving integrators for mechanical systems with explicit time-dependent forcing. Finally, we consider 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.