A Theoretical and Numerical Study of the Dzhanibekov and Tennis Racket Phenomena

Abstract

In this paper, we present complete explanation of the Dzhanibekov phenomenon demonstrated in a space station (www.youtube.com/watch?v=L2o9eBl_Gzw) and the tennis racket phenomenon (www.youtube.com/watch?v=4dqCQqI-Gis). These phenomena are described by Euler’s equation of an unconstrained rigid body that has three distinct values of moments of inertia. In the two phenomena, the rotations of a body about the principal axes that correspond to the largest and the smallest moments of inertia are stable. However, the rotation about the axis corresponding to the intermediate principal moment of inertia becomes unstable, leading to the unexpected rotations that are the basis of the phenomena. If this unexpected rotation is not explained from a complete perspective which accounts for the relevant physical and mathematical aspects, one might misconstrue the phenomena as a violation of the conservation of angular momenta. To address this, especially for students, we investigate the phenomena using more precise mathematical and graphical tools than those employed previously. Following Élie Cartan [1], we explicitly write the vector basis of a body-attached, moving coordinate system. Using this moving frame method, we describe the Newton and Euler equations. The adoption of the moving coordinate frame expresses the rotation of the body more clearly and allows us to use the Lie group theory of special orthogonal group SO(3). We integrate the torque-free Euler equation using the fourth-order Runge-Kutta method. Then we apply a recovery equation to obtain the rotation matrix for the body. By combining the geometrical solutions with numerical simulations, we demonstrate that the unexpected rotations observed in the Dzhanibekov and the tennis racket experiments preserve the conservation of angular momentum.