The goal of this informal working seminar is to understand classical integrable systems described by both ODEs and PDEs. The focus will be on the mechanics of computing explicit solutions to integrable systems using techniques from complex analysis and the theory of Riemann surfaces. The overarching goal at first will be discussing how old approaches to solve completely integrable systems from classical mechanics that take the form of ODEs relate to more recent approach to produce solutions to completely integrable PDEs such as the KdV equation.
In the first meeting we will focus on the derivation of elliptic solutions to the spherical pendulum by enforcing conservation laws. This will be based on results from the chapters on elliptic functions from Markushevich’s “Theory of Functions of a Complex Variable.” There is a connection between the elliptic function solution to the spherical pendulum and the traveling wave solutions to the KdV equation that will be explained.