Simulating the Action Principle in Optics

Light has a fascinating property: it always travels the path that takes the least time between any two points. This is the motivating property behind optical phenomena such as Reflection and Refraction. The unreasonable economic efficiency of light is captured by a single proposition: the Principle of Least Action (PLA) in Optics. The PLA can be difficult for students to comprehend because unlike reflection and refraction, which emerge from optimizing a one-dimensional function, the PLA emerges from optimizing an infinite-dimensional functional. Furthermore, students may be confused by the formulation of the Lagrangian, or the counterintuitive paths generated by the PLA. To address these difficulties, this paper presents three simulations which demonstrate the PLA in Optics. Simulations have proven to be effective pedagogical tools in making mathematical abstractions such as the Action accessible to undergraduate students. The three simulations in this paper are as follows: light reflecting in equal angles, light refracting through different mediums, and light traveling the path of least time in an arbitrary refractive profile. We then consider the famous Brachistochrone problem as a special case of the third simulation. The interactive simulations discussed in the paper, along with supplemental discussion questions, video tutorials, and fully commented code, are available here.