proof

(This is a writeup of Exercise 1.3 from Sussman and Wisdom’s "Structure and Interpretation of Classical Mechanics". See the solutions repository for more.) Law of Reflection Geometry Calculus Law of Refraction Calculus Geometry The problem explores some consequences for optics of the principle of least time. The exercise states: Fermat observed that the laws of reflection and refraction could be accounted for by the following facts: Light travels in a straight line in any particular medium with a velocity that depends upon the medium. The path taken by a ray from a source to a destination through any sequence of media is a path of least total time, compared to neighboring paths. Show that these facts imply the laws of reflection and refraction. ...

I’ve been reading the lovely Visual Complex Analysis by Tristan Needham, and the visual-style proofs he’s been throwing down have been wonderful and refreshing. I’ll write more about this book and its goals later, but I was inspired this AM to write up a proof of the half angle identities from trigonometry using some of the tools from the book. Here’s the half angle identity for cosine: This is an equation that lets you express the cosine for half of some angle in terms of the cosine of the angle itself. As you can imagine, there are double-angle, triple angle, all sorts of identities that you can sweat out next time you find yourself in a 9th grade classroom. ...