  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.
