Euler--Poincar\'e reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

TL;DR

This paper develops a discrete Euler--Poincaré reduction framework for systems on Lie groups with advected parameters, extending Kelvin--Noether theorems and demonstrating its application to underwater vehicle dynamics with numerical validation.

Contribution

It introduces a novel discrete reduction method using group difference maps, extending Kelvin--Noether theorems to discrete systems with advected parameters and additional dynamics.

Findings

The scheme preserves geometric properties over long simulations.

Numerical results validate the effectiveness for underwater vehicle dynamics.

Extension of Kelvin--Noether theorems to discrete systems with advected parameters.

Abstract

The Euler--Poincar\'e equations, firstly introduced by Henri Poincar\'e in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and fluid dynamics. These equations have been extended to various settings, such as semidirect products, advected parameters, and field theory, and have been widely applied to mechanics and physics. In this paper, we introduce the discrete Euler--Poincar\'e reduction for discrete Lagrangian systems on Lie groups with advected parameters and additional dynamics, utilizing the group difference map technique. Specifically, the group difference map is defined using either the Cayley transform or the matrix exponential. The continuous and discrete Kelvin--Noether theorems are extended accordingly, that account for Kelvin--Noether quantities of the corresponding continuous and discrete Euler--Poincar\'e equations. As an application, we show both continuous and discrete Euler--Poincar\'e formulations about the dynamics of underwater vehicles, followed by numerical simulations. Numerical results illustrate the scheme's ability to preserve geometric properties over extended time intervals, highlighting its potential for practical applications in the control and navigation of underwater vehicles, as well as in other domains.