Adaptive Variational Integrators

It is now well known that symplectic integrators lose many of their desirable properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators is required. This implies that the adaptive symplectic integrators resulting from applying a symplectic integrator to the transformed Hamilton’s equations are best understood as coming from a type II or type III generating function, as opposed to a type I generating function. In addition, error analysis results and numerical examples are presented.