On two Abelian Groups Related to the Galois Top

TL;DR

This paper explores algebraic structures related to the Galois top in mathematical physics, focusing on invariants and their connection to the Huygens-Steiner theorem.

Contribution

It defines an abelian semigroup and an abelian group linked to the Galois axis and the application of the Huygens-Steiner theorem.

Findings

Identifies two algebraic motion invariants of the Galois top.

Introduces a new algebraic framework involving an abelian semigroup and group.

Connects invariants to the Huygens-Steiner theorem in rigid body dynamics.

Abstract

In mathematical physics the Galois top, introduced by S. Adlaj, possesses a fixed point on one of two Galois axes through its center of mass. This heavy top has two algebraic motion invariants and an additional transcendental motion-invariant. This third invariant depends on an antiderivative of a variable in the canonical phase space. In this article an abelian semigroup and an abelian group are defined that are related to the application of the Huygens-Steiner theorem to points on the Galois axis of a rigid body.