Reduction and reconstruction aspects of second-order dynamical systems with symmetry

TL;DR

This paper explores how second-order dynamical systems with symmetry can be reduced and reconstructed using connection theory, providing insights into their decomposition and integral curve reconstruction.

Contribution

It introduces a method to decompose and reconstruct second-order systems with symmetry using connection theory, enhancing understanding of their structure.

Findings

Decomposition of vector fields into three parts due to symmetry

Reconstruction of original integral curves from reduced dynamics

Illustrative example confirming theoretical results

Abstract

We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results.