Arithmetic Euler Top

TL;DR

This paper develops an arithmetic analogue of the Euler equations for rigid body dynamics, replacing derivatives with Fermat quotients, extending classical differential equations into an arithmetic framework.

Contribution

It introduces the first arithmetic analogue of the Euler equations for rigid body motion, bridging classical differential equations and number theory.

Findings

Established an arithmetic version of Euler equations

Connected differential equations with Fermat quotients

Extended classical dynamics into arithmetic setting

Abstract

The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlev\'{e}, etc.) were previously shown to possess arithmetic analogues. The paper introduces an arithmetic analogue of the Euler differential equations for the rigid body.