Introduction to Geometric Mechanics and AVI

The goal of this short presentation is to introduce Geometric Mechanics as well as Asynchronous Variational Integrators (AVI). The geometric point of view in mechanics combined with solid analysis has been a phenomenal success in linking various diverse areas, both within and across standard disciplinary lines. For example, symmetry was already widely used in mechanics by the founders of the subject, in such diverse phenomena as reduction, stability, bifurcation... Mechanics has two main points of view, that we will present : Lagrangian mechanics and Hamiltonian mechanics. Lagrangian mechanics is more fundamental, since it is based on variational principles. In another sense, Hamiltonian mechanics is more fundamental, since it is based directly on the energy concept. Fortunately, in many case these branches are equivalent. Many numerical integrators for mechanical system simulation are created by using discrete algorithms to approximate the continuous equations of motion. In this introduction, we present briefly a procedure to construct time-stepping algorithms that approximate the flow of continuous ODE's for mechanical systems by discretizing Hamilton's principle rather than the equations of motion. In particular we will present Asynchronous Variational Integrator (AVI) and its excellent accuracy, conservation and convergence characteristics.