Optimization problems in geometry and physics (NWI-WP032)

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Course info on Osiris (here as pdf)

Lecture: Tue, 10:30–12:15 in HG00.062 (3rd quarter)
Tutorial: Thu, 8:30–10:15 in HG00.108 (3rd quarter)

Exam 1: Fri, March 29, 2019, 8:30–10:30, HG00.068
Exam 2: Thu, June 27, 2019, 8:30–10:30, HG00.068

Contact details:
Lecturer: Dr. Annegret Burtscher, HG03.744
Teaching assistants: Arwin Kool, Matthijs Vernooij

Office hours: Tue, 13:30–14:30 in my office, or by appointment

Course description: 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 may be discussed briefly.

Lecture notes and textbooks: Some lecture notes will be provided, but you are advised to also take notes during lectures. The following textbooks can be useful in addition:
– Hansjörg Kielhöfer, Calculus of Variations, Springer, 2018. ISBN: 978-3-319-71123-2
– Bruce van Brunt, The Calculus of Variations, Springer, 2004. ISBN 978-0-387-21697-3

Prerequisites: A solid understanding of linear algebra and calculus is required. Basic knowledge of analysis is recommended. Knowledge of ordinary differential equations is useful.

Assignments and tutorial: New problems will be assigned each week. You will try these problems at home yourself first, and then get the opportunity to discuss them with fellow students and the teaching assistant during the tutorial. [More information will be provided later.]

Academic Integrity: You will work cooperatively on the problems and are also encouraged to discuss the assignments with your fellow students and the teaching assistant. The solutions, however, have to be written up independently by each student and must be expressed in her/his own words. Citations must be provided for external input such as books, web pages etc.

Grading: The final grade will be based on a weighted average of the homework and the final exam as follows:

  • Homework ….. 15%
  • Final exam ….. 85%

Please contact me if you have any questions.