Illustrations of rigid body motion along a separatrix in the case of Euler-Poinsot

Abstract

The aim of our paper is to explain a computer animation of the strictly critical rigid body motion, which ought not be confused with any other motion in its “proximity”, however close. We demonstrate that the (local) “uniqueness theorem” remarkably fails in the case of critical motion which (time) domain must be compactified via adjoining the point at (complex) infinity. Two (opposite to each other) “flips” correspond to one and the same (initial) rotation, exclusively either clockwise or counterclockwise, (strictly) about the intermediate axis of inertia. These two symmetrical reversals of the direction of the intermediate axis (of inertia), initially matching then opposing the direction of the (fixed) angular momentum, share one and the same (symmetry) axis, which we call “Galois axis”. The Galois axis, which is fixed within the body (but coincides with no principal axis of inertia), rotates uniformly in a plane orthogonal to the angular momentum, as our animation demonstrates. The animation also traces the corresponding two (recurrently self-intersecting) herpolhodes, which turn out to be mirror-symmetrical. The “mirror” is exhibited to lie in a plane, orthogonal to Galois axis at the midst of the “flip”. The Galois axis itself is reflected across the minor (or the major) axis of inertia if the direction of the angular momentum is reversed. The formula for the “swing” of the intermediate axis in the plane orthogonal to Galois axis (in body’s frame), turns out to be “a square root” of Abrarov’s critical solution for a simple pendulum, which (imaginary) period is (exactly) calculated.