Implicit Lagrange-Routh Equations and Dirac Reduction

TL;DR

This paper generalizes Routh's reduction for Lagrangian systems with symmetry, introducing implicit equations derived from the Hamilton-Pontryagin principle and connecting them to Dirac structures for a broader reduction framework.

Contribution

It extends Routh reduction to non-regular Lagrangians using implicit equations and Dirac structures, broadening the applicability of symmetry reduction methods.

Findings

Implicit Lagrange-Routh equations derived from Hamilton-Pontryagin principle.

Reduction of implicit equations using Dirac structures.

Framework applicable to non-regular Lagrangian systems.

Abstract

In this paper, we make a generalization of Routh's reduction method for Lagrangian systems with symmetry to the case where not any regularity condition is imposed on the Lagrangian. First, we show how implicit Lagrange-Routh equations can be obtained from the Hamilton-Pontryagin principle, by making use of an anholonomic frame, and how these equations can be reduced. To do this, we keep the momentum constraint implicit throughout and we make use of a Routhian function defined on a certain submanifold of the Pontryagin bundle. Then, we show how the reduced implicit Lagrange-Routh equations can be described in the context of dynamical systems associated to Dirac structures, in which we fully utilize a symmetry reduction procedure for implicit Hamiltonian systems with symmetry.