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.
Highlights
A comparative investigation of analytic representations of the solution of the dynamic Euler equations, for the inertial motion of a triaxial rigid body about an axis possessing an intermediate value of the moment of inertia and passing through the center of mass, is performed
The solutions have been exhibited to be obtained one from another via corresponding symmetry transformations. They correspond to opposite initial rotations of the rigid body, as well as, opposite directions of “flips” in the process of intermediate axis direction reversal
It rotates uniformly about the angular momentum vector, even as the intermediate axis of inertia reverses its direction with respect to the that vector
Summary
A comparative investigation of analytic representations of the solution of the dynamic Euler equations, for the inertial motion of a triaxial rigid body about an axis possessing an intermediate value of the moment of inertia and passing through the center of mass, is performed. The attainment of an exact and numerically stable solution of the problem of the rigid body motion along the separatrix was achieved via constructing a fourth axis (distinct from the three main axes of inertia), which is called Galois critical axis [4, 5]. It (and only it) rotates uniformly (that is, at a constant angular velocity) about the angular momentum vector, even as the intermediate axis of inertia reverses its direction with respect to the that (constant) vector. One might show that in order to obtain the Galois axis, one ought to rotate the minor axis, in its “initial position”, clockwise for an angle arccos c, looking from the end of the vector of the angular momentum
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