A "No-Go" Theorem for the Existence of an Action Principle for Discrete Invertible Dynamical Systems

TL;DR

This paper proves that a least-action principle cannot exist for finite configuration space invertible discrete dynamical systems, but can be constructed for some infinite configuration space systems through phase space restriction.

Contribution

It establishes a no-go theorem for finite systems and identifies conditions under which an action principle can be recovered in infinite systems.

Findings

No least-action principle exists for finite configuration space systems.

Some infinite systems admit a constructed least-action principle.

Examples illustrate the applicability and limitations of the theorem.

Abstract

In this paper we study the problem of the existence of a least-action principle for invertible, second-order dynamical systems, discrete in time and space. We show that, when the configuration space is finite, a least-action principle does not exist for such systems. We dichotomize discrete dynamical systems with infinite configuration spaces into those of finite type for which this theorem continues to hold, and those not of finite type for which it is possible to construct a least-action principle. We also show how to recover an action by restriction of the phase space of certain second-order discrete dynamical systems. We provide numerous examples to illustrate each of these results.