Hill's equation, tire tracks and rolling cones

Abstract

Louis Poinsot has shown in 1854 that the motion of a rigid body, with one of its points fixed, can be described as the rolling without slipping of one cone, the 'body cone', along another, the 'space cone', with their common vertex at the fixed point. This description has been further refined by the second author in 1996, relating the geodesic curvatures of the spherical curves formed by intersecting the cones with the unit sphere in Euclidean $\mathbb{R}^3$, thus enabling a reconstruction of the motion of the body from knowledge of the space cone together with the (time dependent) magnitude of the angular velocity vector. In this article we show that a similar description exists for a time dependent family of unimodular $ 2 \times 2 $ matrices in terms of rolling cones in 3-dimensional Minkowski space $\mathbb{R}^{2,1}$ and the associated 'pseudo spherical' curves, in either the hyperbolic plane $H^2$ or its Lorentzian analog $H^{1,1}$. In particular, this yields an apparently new geometric interpretation of Schr\"odinger's (or Hill's) equation $ \ddot x + q(t) x =0 $ in terms of rolling without slipping of curves in the hyperbolic plane.