An auxiliary vector space that simplifies some calculations of the Kovalevskaya top

Abstract

We discuss here Kovalevskaya's integrable case of a rigid body, with a symmetric inertia moment, with half the value for the different of inertia moment, and the position of the center of mass and fixed point both placed on a plane orthogonal to the axis of symmetry. We introduce an auxiliary vector space that is a function of two complex conjugate variables and enables ys to simplify many of the calculations necessary to separate the variables in the explicit solution of the Kovalevskaya top. This vector space plays an important role in the study of the elliptic integrals and in particular, in the use Kovalevskaya made of the theory of elliptic integrals.