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Wintersemester 2013/2014

Calculus of Variations

Gerasim Kokarev (Mathematisches Institut der LMU)

"La nature agit toujour par les voies les plus courtes." Fermat, 1662.

Course description:
The course is an introduction to the calculus of variations, the subject concerned with the construction of optimal shapes, states, and processes. The problems of finding optimal solutions and describing their properties played central role during the whole histrory of science. Such problems occur in many questions in physics, engeneering, and economics (as well as daily life) where one regularly has to decide which solution is best or worst, which object has some property to a highest or lowest degree, what is the optimal strategy to reach some goal.

The course is oriented on students in Mathematics and Physics, and is one of the modules in the Mathematics Master Programme as well as the Master Programme in Theoretical and Mathematical Physics (TMP). It is especially suitable for students specialising in analysis, geometry, or mathematical physics. The course is worth 9 ECTS points.

Basic modules on analysis and differential geometry. The course can be taken simultaneously with the module "Differenzierbare Mannigfaltigkeiten/Differential geometry".

Lectures schedule:
Lectures will be given in English twice a week; 10.00-12.00 Tue, Room B004, and 10.00-12.00 Thu, Room B 045.

Course outline:
The course starts with a detailed exposition of the classical theory, covering such topics as first and second variations, symmetries and conservation laws, elements of Hamilton-Jacobi theory. We also plan to discuss isoperimetric problems and Lagrange's multiplier rule, elements of convex analysis and optimal control. Later we study harmonic maps -- a specific variational problem for maps between Riemannian manifolds (known also as a sigma-model in the physics of elementary particles), important in geometry and physics.

Reading list:
1. van Brunt, B. The calculus of variations. Universitext. Springer-Verlag, New York, 2004. xiv+290 pp.
2. Giaquinta, M., Hildebrandt, S. Calculus of variations. I. The Lagrangian formalism. Grundlehren der Mathematischen Wissenschaften, 310. Springer-Verlag, Berlin, 1996. xxx+474 pp.
3. Hélein, F. Harmonic maps, conservation laws and moving frames. Translated from the 1996 French original. With a foreword by James Eells. Second edition. Cambridge Tracts in Mathematics, 150. Cambridge University Press, Cambridge, 2002. xxvi+264 pp.
4. Jost, J. Two-dimensional geometric variational problems. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1991. x+236 pp.

Exercise classes (Übungen):
The exercise classes will hold at 14.00-16.00 Fri, Room B132.

The problem sets for the exercise classes are posted on this web-page on Fridays and are due a week later by Friday 12am. Students are encouraged to do as many problems in these sets as possible, since they comprise an important part of the course and similar problems are likely to appear on the exam.

Problem Set 1 Problem Set 2 Problem Set 3 Problem Set 4
Problem Set 5 Problem Set 6 Problem Set 7 Problem Set 8
Problem Set 9 Problem Set 10 Problem Set 11 Problem Set 12

Final Exam:
There will be an oral exam at the end of February. If you would like to take the exam please send an email to register and arrange the date and time. The registration deadline is 17 January 2014.

Course Programme:

1. Introduction: classical variational problems

1.0. Motivation and examples from geometry and physics (Hamilton's principle, catenary, brachystochrone, geodesics and minimal surfaces).
1.1. Classical variational problems; first variation and Euler-Lagrange equations, elementary regularity theorems.
1.2. Conservation laws, symmetries and Noether's theorem.
1.3. Free boundary conditions, Bolza variational problem, classical isoperimetric problem and Lagrange multipliers, solution to Dido's isoperimetric problem.
1.4. Second variation I: Legendre-Hadamard and Weierstrass necessary conditions; weak and strong extrema.
1.5. Second variation II: sufficient conditions for the extrema based on convexity, Jacobi operator, and the Jacobi theory of conjugate points; solution to the brachystochrone problem.
1.6. Elements of the Hamilton-Jacobi theory: Legendre transform and canonical equations, symplectomorphisms and their generating functions, Hamilton-Jacobi equation, integrability.

2. General variational problems

2.1. Elements of functional analysis: Lagrange variations, Gateaux and Frechet derivatives, Lyusternik theorem, tangent space theorem.
2.2. Lagrange multipliers principle in Banach spaces and its applications; general Lagrange problem and variational problems with higher derivatives.
2.3. Elements of optimal control and Pontryagin's maximum principle, illustrating examples; solution to Newton's aerodynamic problem.

3. Harmonic maps between Riemannian manifolds

3.1. Variational problems with multiple integrals; holonomic constraints.
3.2. Setting on Riemannian manifolds: the Dirichlet integral, the Laplace-Beltrami operator, the second fundamental form, totally geodesic maps.
3.3. The harmonic map equation: extrinsic and intrinsic forms of the equation, conservation laws and their consequences.
3.4. Basic properties of harmonic maps: unique continuation, maximum principles, rank one theorem, maps into totally geodesic submanifolds.
3.5. Short survey on the existence of harmonic maps; the Bochner formula and its applications.
3.6. Harmonic maps of surfaces: Rado-Kneser-Lewy theorem, Hopf differential and its properties, branch points, Riemann-Hurwitz formula, non-existence of harmonic maps.

Last modified: 23 Sep 2013