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It is well known that symplectic integrators lose their near energypreservation properties when variable step sizes are used. The most commonapproach to combine adaptive step sizes and symplectic integrators involves thePoincar\'e transformation of the original Hamiltonian. In this article, weprovide a framework for the construction of variational integrators using thePoincar\'e transformation. Since the transformed Hamiltonian is typicallydegenerate, the use of Hamiltonian variational integrators based on Type II orType III generating functions is required instead of the more traditionalLagrangian variational integrators based on Type I generating functions. Erroranalysis is provided and numerical tests based on the Taylor variationalintegrator approach of Schmitt, Shingel, Leok (2018) to time-adaptivevariational integration of Kepler's 2-Body problem are presented. Finally, weuse our adaptive framework together with the variational approach toaccelerated optimization presented in Wibisono, Wilson, Jordan (2016) to designefficient variational and non-variational explicit integrators for symplecticaccelerated optimization.

Adaptive Hamiltonian Variational Integrators and Symplectic Accelerated Optimization