Mats Vermeeren (TU Berlin, Germany)
Modified equations for variational integrators
Numerical discretizations of differential equations are often studied through their modified equation. This is a perturbation of the original differential equation with solutions that interpolate the discretization. It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then its modified equation is again Hamiltonian. This explains why symplectic integrators very nearly conserve energy. In this talk we discuss this property from the Lagrangian side. We present a technique to construct a Lagrangian for the modified equation of a variational integrator. This is particularly interesting in the case of degenerate Lagrangians, where the Legendre transform cannot be used to switch between the Lagrangian and Hamiltonian formulations.