  A fixed time-step variational integrator cannot preserve momentum, energy,
and symplectic form simultaneously for nonintegrable systems. This barrier can
be overcome by treating time as a discrete dynamic variable and deriving
adaptive time-step variational integrators that conserve the energy in addition
to being symplectic and momentum-preserving. Their utility, however, is still
an open question due to the numerical difficulties associated with solving the
discrete governing equations. In this work, we investigate the numerical
performance of energy-preserving, adaptive time-step variational integrators.
First, we compare the time adaptation and energy performance of the
energy-preserving adaptive algorithm with the adaptive variational integrator
for Kepler's two-body problem. Second, we apply tools from Lagrangian backward
error analysis to investigate numerical stability of the energy-preserving
adaptive algorithm. Finally, we consider a simple mechanical system example to
illustrate the backward stability of this energy-preserving, adaptive time-step
variational integrator.
