Un-Reduction of Systems of Second-Order Ordinary Differential Equations

TL;DR

This paper introduces a novel approach to un-reduction for second-order ODE systems, establishing a primary un-reduced SODE and exploring its relations and applications beyond traditional Lagrangian frameworks.

Contribution

It defines a primary un-reduced SODE and demonstrates how all other un-reduced SODEs relate to it, extending un-reduction techniques beyond Lagrangian systems.

Findings

Introduction of a primary un-reduced SODE

Relations among un-reduced SODEs established

Applications beyond classical Lagrangian systems

Abstract

In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential equations (SODEs), one may associate to the first system a (what we have called) "primary un-reduced SODE", and we explain how all other un-reduced SODEs relate to it. We give examples that show that the considered procedure exceeds the realm of Lagrangian systems and that relate our results to those in the literature.