A note on relative equilibria of multidimensional rigid body

TL;DR

This paper explores the conditions under which rotations of multidimensional rigid bodies are stationary, extending known results from three dimensions and analyzing cases with coinciding eigenvalues of the angular velocity matrix.

Contribution

It provides a detailed description of stationary rotations in multidimensional rigid bodies, especially when eigenvalues of the angular velocity matrix coincide, which was not fully understood before.

Findings

Stationary rotations occur when rotations are in principal axes of inertia with distinct eigenvalues.

New characterization of rotations with coinciding eigenvalues of angular velocity matrix.

Extension of classical three-dimensional results to higher dimensions.

Abstract

It is well known that a rotation of a free generic three-dimensional rigid body is stationary if and only if it is a rotation around one of three principal axes of inertia. As it was noted by many authors, the analogous result is true for a multidimensional body: a rotation is stationary if and only if it is a rotation in the principal axes of inertia, provided that the eigenvalues of the angular velocity matrix are pairwise distinct. However, if some eigenvalues of the angular velocity matrix of a stationary rotation coincide, then it is possible that this rotation has a different nature. A description of such rotations is given in the present paper.