Direct Methods in the Calculus of Variations
This course aims to introduce students with mixed mathematical backgrounds to the theory of calculus of variations and its importance in solving problems of geometric nature. To make the course accessible, we will focus on the main ideas while keeping the technical mathematical details under the rug.
Instructor: Shah Faisal, Berlin Mathematical School, Berlin
, Berlin Mematsearch Centre, GermanyIf you have any questionsease write to me at shah.faisal.math@gmail.com
Lecturer 03: Applications to differential geometry-I: Existence of closed geodesics
Abstract: I want to give you a few simple applications of the calculus of variations in differential geometry. Since most of you don't have a background in differential geometry, I will explain some basics of differential geometry. In particular, I will cover the following topic while not going into too much detail:
Smooth manifolds, tangent bundle, Riemannian metric.
What are affine connections, and why do we care about them?
Closed geodesics, and their characterization as the critical points of the energy functional on the loop space.
Date: Jan. 7th, 2023
Book for further reading on these topics: John M. Lee, Introduction to Smooth Manifolds
Here is a Video Recording of this lecture.
Lecturer 02: Gamma-perturbation of functionals
Abstract: In the previous lecture, I explained two techniques for studying the existence and computing minimizers of functionals: a classical method based on solving the Euler–Lagrange equation coming from the first variation of the functional, and a modern approach based on extracting convergent sub-sequences from minimizing sequences of the functional under consideration.
In this lecture, I will explain another direct technique: instead of looking at the original functional, one can look at a nice perturbation of the original functional and extract information about the minimizers of the original functional by studying the perturbed functional. Such a perturbation is known as a Gamma-perturbation. My plan is to explain the notion of Gamma-perturbation with examples and demonstrate with examples how one can recover some minimizers of a given functional via its gamma-perturbations.
Book for further reading on these topics: Gianni Dai Maso, An Introduction to Gamma-Convergence
Here are the details SLIDES and Video Recording from Lecture 02.
Lecture 01: Direct and indirect methods in the calculus of variations
Abstract: In this talk, I introduce the audience to a beautiful piece of Mathematics, the Calculus of Variations. The Calculus of Variations deals with minimizing functionals (possibly non-linear) defined on Banach spaces (possibly infinite-dimensional). To motivate the theory, I start with real-life examples to show how solutions to specific problems in life boil down to minimizing certain real-valued functionals defined on Banach spaces. Using the knowledge of the audience from the undergraduate analysis, I then in a heuristic way develop techniques that help to investigate the existence and computability of minimizers of a class of functionals. I explain both classical and direct methods (modern) to minimize functionals.
Here are the details SLIDES from Lecture 01.
Homework: Here is homework that you can try. Please.
Book for further reading on these topics: Direct Methods in the Calculus of Variations | Bernard Dacorogna | download (b-ok.cc)