The toy top, an integrable system of rigid body dynamics

Abstract

A toy top is defined as a rotationally symmetric body moving in a constant gravitational field while one point on the symmetry axis is constrained to stay in a horizontal plane. It is an integrable system similar to the Lagrange top. Euler-Poisson equations are derived. Following Felix Klein, the special unitary group $\mathrm{SU}(2)$ is used as configuration space and the solution is given in terms of hyperelliptic integrals. The curve traced by the point moving in the horizontal plane is analyzed, and a qualitative classification is achieved. The cases in which the hyperelliptic integrals degenerate to elliptic ones are found and the corresponding solutions are given in terms of Weierstrass elliptic functions.

References in corpus (1)

• Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top

Cited by in corpus (1)

• Motion of the heavy symmetric top when magnitudes of conserved angular momenta are different