Modified equations for variational integrators applied to Lagrangians linear in velocities

TL;DR

This paper develops a framework for deriving modified equations for variational integrators applied to degenerate Lagrangians linear in velocities, including the construction of a Lagrangian for the full system of modified equations.

Contribution

It introduces a method to construct the full system of modified equations for degenerate Lagrangians, extending the Lagrangian structure to these modified equations.

Findings

A Lagrangian for the principal modified equation can be constructed.

The full system of modified equations can be made Lagrangian by doubling the system dimension.

The extended discrete system's Lagrangian leads to a Lagrangian for the full modified system.

Abstract

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.