Modified Equations for Variational Integrators

TL;DR

This paper explores the relationship between variational integrators and their modified equations, providing a method to derive a Lagrangian for the modified equation from the discrete Lagrangian.

Contribution

It introduces a technique to construct a Lagrangian for the modified equation directly from the discrete Lagrangian of a variational integrator.

Findings

The modified equation retains a Hamiltonian structure.

A systematic method to derive the Lagrangian of the modified equation is proposed.

The approach enhances understanding of the geometric properties of variational integrators.

Abstract

It is well-known that if a symplectic integrator is applied to a Hamiltonian system, then the modified equation, whose solutions interpolate the numerical solutions, is again Hamiltonian. We investigate this property from the variational side. We present a technique to construct a Lagrangian for the modified equation from the discrete Lagrangian of a variational integrator.