ILLUSTRATIONS OF RIGID BODY MOTION ALONG A SEPARATRIX IN THE CASE OF EULER-POINSOT

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.


INTRODUCTION
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. Numerical integration of the kinematic equations of the spherical motion of a rigid body is carried out via employing quaternion representation. Four solutions presented in the literature have been exhibited to possess the property of stationary rotation of a special axis, rigidly attached to the body, about the vector of the angular momentum. This axis (the Galois axis) lies in the ALGORITHMIC MATHEMATICS AND MATHEMATICAL MODELING plane perpendicular to the intermediate axis of inertia and its position is determined via the ratios of the principal values of the moments of inertia. 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.

SOLUTIONS OF THE DYNAMIC EULER-POINSOT EQUATIONS FOR MOTION ALONG A SEPARATRIX
A known classical result in rigid body dynamics is the description of motion via a geometric interpretation, in which the body motion is associated with the motion of a point along curves of the intersection of the surface of constant energy with the sphere constructed in the coordinates of the angular momentum. Generally, when m 2 2hB , these are сlosed spatial algebraic curves of the fourth order. In the case of m 2 = 2hB they are intersecting circles. For A = 11, B = 21, C = 32, m/B = 1 these separatrices are represented in Fig. 1. are expressed explicitly via Jacobi elliptic functions [9]. Alternatively, they might be expressed via the alternative elliptic function, as presented in [3,5]. If m 2 = 2hB , then the rigid body motion is along a separatrix and the solution is written in terms of hyperbolic functions. This coinciding form of solution was repeatedly reproduced in the scientific and educational literature. In particular, following the works of [10,11], the solution can be represented in the following form: The works [8,12] are confined to examining only the first of these solutions.
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. The four solutions thus obtained have the form:  (3) and (4), while distinct in form, are equivalent and their apparent difference is due to the chosen ordering of the moments of inertia A, B and C , either ascendingly or descendingly. However, the choice of ordering is elusive in the general case, where m 2 2hB , and one ought (in fact) adhere to the ordering imposed via the rule given in [4,5]. In the invariant description of the critical motion, the systems (3) and (4) contain two pairs of identical solutions, which are due to two possible initial positions of the angular momentum with respect to the chosen (inertial) coordinate system. One might also show that in order to obtain the Galois axis, one ought to rotate the minor axis, 1 in its "initial position", clockwise for an angle arccos c, looking from the end of the vector of the angular momentum. 1 We risk no confusion in our paper as the minor axis of McCullagh ellipsoid (which we employ) does coincide with the principal axis, corresponding to the smallest moment of inertia.

INTEGRATION OF KINEMATIC EQUATIONS
In order to describe the motion of the body, we use the quaternion parametrization, which allows us to write down the relationship between the coordinates of the unit quaternion and the projections of the angular velocity vector on the axis of the moving coordinate system in the form of a system of linear homogeneous differential equations [7]: The results of integrating these equations on the interval −8π ≤ t ≤ 8π for the initial condi-  The corresponding screenshot of the program is shown in Fig. 3.
The change in the angular velocity projections onto the moving axes according to the first equation of system (4), for the given parameters, is demonstrated in Fig. 4.

ILLUSTRATION OF THE MOTION
The results of integrating the kinematic equations make possible the formation of a functional rotation matrix [6] as well as, the description of the motion of each point of the rotating body in fixed axes by means of the matrix algorithm In full, the motion is illustrated by a synchronous animation https://www.youtube.com/ watch?v=c0m_yeKeCiQ. An invariant pair of synchronous solutions is represented by the following synchronous animation https://www.youtube.com/watch?v=e9wGPh-iiRw.

CONCLUSIONS
The animation of the critical rigid body motion along a separatrix was posted on Utube (https://www.youtube.com/watch?v=e9wGPh-iiRw) on February 14, 2018. Ten days later, the first comment was posted by Francois Lamarche (McGill University): "very nice indeed, really shows how it keeps rotating with the same angular momentum, but in a completely opposite direction". Two (synchronously) displayed solutions correspond to one and the same initial rotation (exclusively either "clockwise" or "counterclockwise"). Both solutions share one and the same (fixed within the body) uniformly rotating Galois axis, as further clarified in [5]. We emphasize that the law of motion, governing the swing (in a semicircle) of the intermediate axis of inertia in the plane, orthogonal to Galois' axis (in body's frame of reference), is precisely a "square root" of Abrarov's critical motion for a simple pendulum [1][2][3]. The imaginary period of such critical motion matches (in modulus) 2πB /(βm) = 2π ABC / 2h(C − B )(B − A).