Canonical Form of the Inertia Tensor for Rigid Bodies on the Lobachevskii Plane and in a Pseudo-Euclidean Space

Abstract

This paper studies the properties of the inertia tensor for tops in 3-dimensional (pseudo-) Euclidean space and for plates on the Lobachevskii plane. A canonical form is described to which one can reduce the (pseudo-)Euclidean inner product and the inertia operator of any top by changing the basis. All tops whose inertia operators are nondiagonalizable (every top of this kind lies in a plane tangent to the isotropic cone) are described. All realizable triples \((J_1,J_2,J_3)\) of principal moments of inertia are described in terms of triangle inequalities and their pseudo-Euclidean analogs. A criterion for the realizability of such a triple by a plate on the Lobachevskii plane is obtained. All realizable pairs of triples \((J_1,J_2,J_3)\) of principal moments of inertia and coordinates of the center of mass \((c_1,c_2,c_3)\) of a top in the principal axes of inertia are described in both the diagonalizable and nondiagonalizable cases. In all cases, it is shown that a body consisting of no more than six points can be taken as the realizing top. Eigenvalues of all degenerate inertia operators are described. As an application, realizable examples of integrable tops in 3-dimensional pseudo-Euclidean space (the Euler and Lagrange tops) are described.