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：本文从两种不同视角出发，研究粒子穿越两种不同势能区域界面时对应的斯涅尔定律（Snell's law）类比问题。首先，展示了一项涉及弹珠与势能阶跃的简易演示实验。随后，从理论视角展开推导：基于莫佩尔蒂（Maupertuis）与雅可比（Jacobi）变分原理，推导出描述力学折射的该定律，其推导逻辑与借助费马最短时间原理（Fermat's least time principle）推导经典斯涅尔折射定律的经典过程高度相似。最后，通过将牛顿色散关系替换为相对论色散关系，并依托莫佩尔蒂变分原理，得到了力学折射的相对论形式。此外，本文还探讨了该教学处理方式的教育意义。