Dirac Reduction for Nonholonomic Mechanical Systems and Semidirect Products

TL;DR

This paper develops a symmetry-based Dirac reduction framework for nonholonomic systems on Lie groups, extending variational and dynamical structures, with applications to complex fluids and infinite-dimensional systems.

Contribution

It introduces a novel Dirac reduction method for nonholonomic systems on Lie groups with broken symmetry, unifying Lagrangian and Hamiltonian reductions.

Findings

Reduced equations include implicit Euler-Poincaré-Suslov and Lie-Poisson-Suslov equations.

Application to Rivlin-Ericksen fluids as infinite-dimensional nonholonomic systems.

Illustrations with finite and infinite dimensional examples.

Abstract

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange-Dirac and Hamilton-Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler-Poincar\'e-Suslov equations with advected parameters and the implicit Lie-Poisson-Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin-Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.