Mechanical refraction in action

Abstract We study the analog of Snell's law for particles moving across the interface of two regions with two different potential energies, from two different points of view. First, a simple demonstration involving marbles and a potential step is shown. Then, from a theoretical point of view, this law describing mechanical refraction is derived from the Maupertuis and Jacobi variational principles, in close analogy with the well known derivation of Snell's law for refraction from Fermat's least time principle. Finally, the relativistic version of mechanical refraction is obtained by the Maupertuis principle, by trading the Newtonian dispersion relation with the relativistic one. The pedagogical significance of this treatment is discussed.摘要 我们从两种不同视角出发，研究粒子跨越两类不同势能区域界面时的斯涅尔定律（Snell's Law）类比问题。首先，我们展示了一项涉及弹珠与势阶的简易演示实验。随后，从理论维度而言，这一力学折射定律可由莫佩尔蒂与雅可比变分原理推导得出，其推导过程与借助费马最短时间原理推导斯涅尔定律的经典范式高度相似。最后，通过将牛顿色散关系替换为相对论色散关系，借助莫佩尔蒂原理得到了力学折射的相对论形式。本文还讨论了该研究处理方式的教学意义。