Modified equations for variational integrators applied to Lagrangians linear in velocities
Provides a theoretical foundation for analyzing numerical methods in mechanics, but is incremental as it extends existing techniques to a specific class of Lagrangians.
This paper extends the theory of modified equations for variational integrators to degenerate Lagrangians linear in velocities, constructing Lagrangians for both the principal and full modified equations via a doubling technique.
Variational integrators applied to degenerate Lagrangians that are linear in the velocities are two-step methods. The system of modified equations for a two-step method consists of the principal modified equation and one additional equation describing parasitic oscillations. We observe that a Lagrangian for the principal modified equation can be constructed using the same technique as in the case of non-degenerate Lagrangians. Furthermore, we construct the full system of modified equations by doubling the dimension of the discrete system in such a way that the principal modified equation of the extended system coincides with the full system of modified equations of the original system. We show that the extended discrete system is Lagrangian, which leads to a construction of a Lagrangian for the full system of modified equations.