Euler's rigid rotators, Jacobi elliptic functions, and the Dzhanibekov or tennis racket effect

Abstract

In this paper, the torque-free rotational motion of a general rigid body is developed analytically and is applied to the flipping motion of a T-handle spinning in zero gravity that can be seen in videos on the internet. This flipping motion is known both as the Dzhanibekov effect (after the cosmonaut who reported it) and more recently the tennis racket effect. The presentation is self-contained, accessible to students, and is complementary to the treatment found in most texts in that it involves a time-dependent analytical solution in terms of elliptic functions as opposed to a development based on conservation laws. These two complementary approaches are interesting and useful in different ways. In the present approach, the Euler rigid-body equations are derived and then solved as differential equations that are satisfied by Jacobi elliptic functions. This is analogous to solving the spring–mass harmonic oscillator problem by turning Newton's laws into differential equations that are satisfied by sine and cosine functions. The Jacobi functions are closely related to these trigonometric functions and are only slightly more complicated. They are defined as geometrical ratios on a reference ellipse and developed geometrically without reference to power series or complex variables. However, because these functions are less familiar, they are introduced in a short Appendix where their main properties are derived. Also, a link is provided to a Mathematica script for animating the analytical solution to the present problem.