On two Abelian Groups Related to the Galois Top
This work addresses a theoretical problem in mathematical physics for researchers studying rigid body dynamics and algebraic invariants, but it appears incremental as it builds on existing concepts like the Galois top and Huygens-Steiner theorem.
The paper tackles the problem of understanding the algebraic structures related to the Galois top in mathematical physics by defining an abelian semigroup and an abelian group connected to the Huygens-Steiner theorem applied to points on the Galois axis of a rigid body, with the result being the introduction of these new algebraic objects without specific numerical outcomes.
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.