The Theory of Caustics and Wavefront Singularities with Physical Applications

TL;DR

This paper reviews Arnold's work on caustics and wavefront singularities, highlighting their mathematical foundations and applications to physics, especially in Hamilton-Jacobi theory and wave propagation.

Contribution

It provides an accessible introduction to the theory of Lagrangian and Legendrian submanifolds and demonstrates their relevance to physical phenomena like caustics and wavefronts.

Findings

Characterization of caustics and wavefront singularities

Application of singularity theory to Hamilton-Jacobi equations

Insights into null surfaces and wave propagation phenomena

Abstract

This is intended as an introduction to and review of the work of V, Arnold and his collaborators on the theory of Lagrangian and Legendrian submanifolds and their associated maps. The theory is illustrated by applications to Hamilton-Jacobi theory and the eikonal equation, with an emphasis on null surfaces and wavefronts and their associated caustics and singularities.