Polite actions of non-compact Lie groups

TL;DR

This paper introduces the concept of polite group actions, a generalization of proper actions, to better understand invariant functions and facilitate reduction and reconstruction in dynamical systems with symmetry groups.

Contribution

It defines polite group actions and demonstrates their applicability in reduction and reconstruction processes in dynamical systems.

Findings

Polite actions generalize proper group actions.

Polite actions enable reduction and reconstruction in dynamical systems.

Invariant functions play a central role in the framework.

Abstract

Based mainly on examples of interest in mechanics, we define the notion of a polite group action. One may view this as not only trying to give a more general notion than properness of a group action, but also to more fully understand the role of invariant functions in describing just about everything of interest in reduction. We show that a polite action of a symmetry group of a dynamical system admits reduction and reconstruction.