Symmetries and Lagrangian time-discretizations of Euler equations

TL;DR

This paper explores symmetry conditions in Lagrangian discretizations of Euler equations on Lie groups, revealing properties of integrable systems and their discretizations.

Contribution

It extends previous work by analyzing symmetry properties of Lagrangian discrete systems on infinite-dimensional Lie groups.

Findings

Lagrangian discrete systems can be integrable time-discretizations of Euler equations.

Symmetry conditions influence the properties of these discretizations.

The paper discusses interesting properties of these systems.

Abstract

In the late 80s - early 90s J. Moser and A. P. Veselov considered Lagrangian discrete systems on Lie groups with additional symmetry conditions imposed on Lagrangians. They observed that such systems are often integrable time-discretizations of integrable Euler equations on these Lie groups. In recent papers we studied Lagrangian discrete systems with additional symmetry requirements on certain infinite-dimensional Lie groups. We will discuss some interesting properties of these systems.