- The student is acquainted with classical one-dimensional optimization problems in geometry and physics that mark the starting point of the calculus of variations. The student knows how to approach and solve these problems using indirect methods.
- The student can reformulate a variational problem to an Euler-Lagrange equation (when possible), and is aware of the functional analytic background required.
- The student can analyze variational problems with and without constraints.
- The students knows that not every optimization problem has a minimizer.
Classical problems in physics and geometry are often concerned with minimizing or maximizing certain functionals. For example, one may be interested in finding the shortest path between two points (geodesic) or the path with fastest decent (brachistochrone), or maximizing the area of a figure bounded by a curve (Dido’s problem). In order to study such a (one-dimensional) variational problem it is reformulated into a so-called Euler-Lagrange equation, and in order to solve the optimization problem it then remains to solve a differential equation. Historically, the existence of minimizers was taken for granted, but Weierstraß' counterexample showed that this need not be the case. More modern “direct methods” prove the existence of minimizers via a concrete functional analytic construction of a minimizing sequence.
In this course we will study a few classical optimization problems in geometry and physics. From a theoretical perspective we will cover the notion of a first variation, the fundamental lemma as well as the derivation, analysis and solution of the Euler-Lagrange equation. Noether’s Theorem and some more advanced methods will be discussed briefly.