The Square Cat

TL;DR

This paper introduces a simple two-dimensional 'cat' model demonstrating body rotation without external forces, illustrating gauge theory concepts, non-commutative operators, and topological invariants in different mechanical frameworks.

Contribution

It provides a novel, simplified example connecting gauge theory, non-commutative operators, and topology in classical mechanics models.

Findings

The 'cat' can rotate with zero angular momentum.

Comparison between Newtonian and Aristotelian mechanics shows analogous roles of momentum and torque.

Identification of a topological invariant in the model.

Abstract

We present a simple, two dimensional example of a "cat" -- a body with zero angular momentum that can rotate itself with no external forces. This model is used to explain why this problem is known to be a gauge theory and to illustrate the importance of non-commutative operators. We will also show a comparison between the free-space "cat" in Newtonian mechanics and the same problem in Aristotelian mechanics at low Reynolds number; this simple example shows the analogy between (angular) momentum in Newtonian mechanics and (torque) force in Aristotelian mechanics. We will end by pointing out a topological invariant common to our model in free space and at low Reynolds number.