Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations

TL;DR

This paper extends the concept of mechanical connections to arbitrary Lagrangians invariant under symmetry groups, deriving new reduced equations and demonstrating solution reconstruction with examples.

Contribution

It generalizes the mechanical connection concept beyond kinetic energy-based Lagrangians and introduces a novel derivation of the Lagrange-Poincare equations.

Findings

Extended mechanical connection to arbitrary Lagrangians.

Derived new form of Lagrange-Poincare equations.

Validated theory with illustrative examples.

Abstract

We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated to Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new fashion and we show how solutions of the Euler-Lagrange equations can be reconstructed with the help of the mechanical connection. Illustrative examples confirm the theory.