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 selfintersecting) 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.

On the second memoir of Évariste Galois' last letter

“…These are the six values β ∈ {±i , ±1/ 2, ± 2}, corresponding to τ = i := −1, and the four values β ∈ {±i ζ, ±i ζ 2 }, corresponding to τ = ζ. 8 An isomorphism between elliptic curves as their elliptic modulus β undergoes permissible transformations (generated by S and T ) might explicitly be given as a linear map between first coordinates. Evidently, the isomorphism corresponding to the transformation β → 1/β is given by the identity map x → x, and the isomorphism corresponding to the transformation β → −β is given by the map x → −x.…”

“…10 7 The latter statement merely defines a modular form of weight zero. 8 A reformulation involving α (instead of β) would be less cumbersome, perhaps, and so we give it here. Generally, six distinct values of α correspond to a single point τ in the fundamental domain.…”

“…the four triple-pairs, which we did write after the first, and we now, guided by his conciseness and brevity, confine ourselves to writing down only the first pair-set that he presented for each of the two remaining cases, where n = 7 and n = 11, respectively: {(0, ∞), (1, 3), (2, 6), (4, 5)} and {(0, ∞), (1,2), (3,6), (4,8), (5,10), (9, 7)}. Unlike the case n = 5, an alternative might be presented for n = 7, which is {(0, ∞), (1,5), (2,3), (4, 6)}, and for n = 11, which is {(0, ∞), (1,6), (3,7), (4,2), (5,8), (9,10)…”

“…In fact, the polynomials r m 5 might be so ordered so that, for each m, the value β 2 m coincides with y 8 m . The (general) expression for y 8 m = β 2 m , as given in (8), might be rewritten for the special case n = 5 as…”

“…One might infer, from equation (8), that the modular polynomial, of level 2, Φ 2 (x, z) vanishes at (x, z l ) = 4 27…”