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On two Abelian Groups Related to the Galois Top

Helmut Ruhland

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.

On two Abelian Groups Related to the Galois Top

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.
Paper Structure (7 sections, 2 theorems, 11 equations)

This paper contains 7 sections, 2 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. An abelian semigroup of inertia maps
  3. An abelian group of maps
  4. A (semi)group extension by the scaling group
  5. An open question
  6. Proof that the image of $j(x)$ is a subset of the codomain
  7. Proof of the (semi)group law

Key Result

Theorem 2.1

The maps $j(d^2)$, where $d$ is the distance of a point $O$ on the Galois axis to the center of mass $G$, define an abelian semigroup $G_s = \{ j(d^2) \, \vert \, d \in \mathbb{R} \} = \{ j(x) \, \vert \, x \in \mathbb{R}_{\geq 0} \}$. $j(0)$ is the neutral element. The semigroup law is $j(x) \circ

Theorems & Definitions (6)

  • Definition 1
  • Theorem 2.1
  • Definition 2
  • Theorem 3.1
  • Definition 3
  • Remark 1