希尔方程，轮胎印和滚动锥
Hill's equation, tire tracks and rolling cones
摘要：
路易斯·波因索在1854年指出，一个刚体的运动，当它的一个点固定时，可以描述为一个圆锥（“体锥”）沿着另一个圆锥（“空间锥”）的滚动而不滑动，它们的共同顶点在不动点。第二作者在1996年进一步完善了这一描述，将圆锥体与欧几里得$\mathbb{R}^3$中的单位球面相交形成的球面曲线的测地曲率联系起来，从而能够根据空间圆锥体的知识以及角速度矢量的（不定常）大小重建物体的运动。在本文中，我们证明了对于不定常的么模$ 2 \times 2 $矩阵族，也存在类似的描述，即3维Minkowski空间中的滚动锥$\mathbb{R}^{2,1}$以及双曲平面$H^2$或其洛伦兹类似物中的相关“伪球形”曲线$H^{1,1}$。特别是，这产生了一个明显的新的几何解释薛定谔（或希尔）方程$ \ddot x + q(t) x =0 $在滚动方面没有滑动的曲线在双曲平面。
英文摘要：
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 Schrödinger's (or Hill's) equation $ \ddot x + q(t) x =0 $ in terms of rolling without slipping of curves in the hyperbolic plane.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
