Quotient Manifold Projections and Hierarchical Dynamics

TL;DR

This paper develops a mathematical framework for hierarchical organization in smooth dynamical systems using quotient manifolds and Lie groups, linking projections to symmetries and invariants.

Contribution

It introduces a novel construction for identifying hierarchical levels via quotient manifolds and Lie groups, connecting projections to system symmetries.

Findings

Hierarchical levels are characterized by quotient manifolds and Lie groups.

Projections to higher levels relate to symmetries of the dynamical system.

Connections are established between quotient projections, invariant manifolds, and Noether's theorem.

Abstract

In this paper we explore the mathematical structure of hierarchical organization in smooth dynamical systems. We start by making precise what we mean by a level in a hierarchy, and how the higher le vels need to respect the dynamics on the lower levels. We derive a mathematical construction for identifying distinct levels in a hierarchical dynamics. The construction is expressed through a quotient manifold of the phase space and a Lie group that fulfills certain requirement with respect to the flow. We show that projections up to higher levels can be related to symmetries of the dynamical system. We also discuss how the quotient manifold projections relate to invariant manifolds, invariants of the motion, and Noether's theorem.