Vladimir Arnold, in memoriam

I missed that Vladimir Arnold has died. Arnold was famous for his own contributions to mathematics, but in my opinion he was also the world’s great expositor of mathematics.

When I first encountered the subject of Lie algebras, I thought it was pointless and unmotivated. I also had the impression from high school physics that classical mechanics was built out of a bunch of random facts that were true for no reason, like the conservation of angular momentum. Also, I thought that potential energy was a sort-of a con — that if you can simply declare that a body has potential energy that you can make the law of conservation of energy tautologically true. Reading Arnold’s Mathematical Methods in Classical Mechanics changed all that. Arnold starts with one-dimensional systems like the inverse-square law and harmonic oscillator, and then to three-dimensional systems where he explains how symmetries in the equations of motion lead to conservation laws. Along the way, he explains how Lie groups lead to Lie algebras, and how in particular how rotational symmetries in 3d lead to the Lie algebra of so(3), which physicists use in the guise of the cross-product of vector calculus. He also introduces the Lagrangian and Hamiltonian formulations of classical mechanics. Most importantly, (since you can learn the equivalent from a physics text like Goldstein’s Classical Mechanics), he puts in the language of mathematicians rather than the language of physicists.

Years after I studied the subject of ODEs, I almost bought Arnold’s (expensive) Ordinary Differential Equations just because it was such a beautiful introduction to the subject. Lots of textbooks allude to the dynamical systems viewpoint for ODEs, but his book really communicates that viewpoint.