Mechanical Snell's Law

TL;DR

This paper explores a mechanical analogy to Snell's law by analyzing the least-time path of a particle constrained to two segments under conservative forces, revealing a relation similar to optical refraction.

Contribution

It introduces a mechanical version of Snell's law for particles moving under conservative forces, linking angles of incidence and refraction to average speeds on path segments.

Findings

The ratio of sines of incidence and refraction angles equals the ratio of average speeds.

The law holds when the horizontal component of the conservative force is zero.

The result extends the analogy between optics and mechanics in constrained motion.

Abstract

We investigate the motion of a massive particle constrained to move along a path consisting of two line segments on a vertical plane under an arbitrary conservative force. By fixing the starting and end points of the track and varying the vertex horizontally, we find the least-time path. We define the angles of incidence and refraction similar to the refraction of a light ray. It is remarkable that the ratio of the sines of these angles is identical to the ratio of the average speeds on the two partial paths as long as the horizontal component of the conservative force vanishes.