  It is well known that symplectic integrators lose their near energy
preservation properties when variable step sizes are used. The most common
approach to combine adaptive step sizes and symplectic integrators involves the
Poincar\'e transformation of the original Hamiltonian. In this article, we
provide a framework for the construction of variational integrators using the
Poincar\'e transformation. Since the transformed Hamiltonian is typically
degenerate, the use of Hamiltonian variational integrators based on Type II or
Type III generating functions is required instead of the more traditional
Lagrangian variational integrators based on Type I generating functions. Error
analysis is provided and numerical tests based on the Taylor variational
integrator approach of Schmitt, Shingel, Leok (2018) to time-adaptive
variational integration of Kepler's 2-Body problem are presented. Finally, we
use our adaptive framework together with the variational approach to
accelerated optimization presented in Wibisono, Wilson, Jordan (2016) to design
efficient variational and non-variational explicit integrators for symplectic
accelerated optimization.

Numerical Analysis;Numerical Analysis