Jointly Equivariant Dynamics for Interacting Particles

TL;DR

This paper characterizes all possible collective particle dynamics that are equivariant under symmetry groups, ensuring independence from coordinate changes, with applications to low-dimensional symmetries in manifold-based systems.

Contribution

It provides a comprehensive classification of jointly equivariant dynamics for interacting particles under symmetry Lie groups, extending understanding of symmetry-invariant collective behaviors.

Findings

Classified all equivariant dynamics for given symmetry groups.

Applied results to low-dimensional symmetries in manifold systems.

Established a framework for symmetry-preserving particle dynamics.

Abstract

Let a finite set of interacting particles be given, together with a symmetry Lie group $G$. Here we describe all possible dynamics that are jointly equivariant with respect to the action of $G$. This is relevant e.g., when one aims to describe collective dynamics that are independent of any coordinate change or external influence. We particularize the results to some key examples, i.e. for the most basic low dimensional symmetries that appear in collective dynamics on manifolds.