Winter 2012

M W F 2:00 - 2:50 JWB 333

Instructor Andrej Cherkaev

Every problem of the calculus of variations has a solution, provided that the word `solution' is suitably understood. David Hilbert

Topics to be covered:

• Basic techniques of Calculus of Variations. Euler equations, Approximation with penalty. Legendre and Weierstrass tests, direct methods.
• Hamiltonian. Mechanical applications. Geometric optics.
• Duality and Legendre transform.
• Variational principles.
• Variational problems for multiple integrals:
• Optimization of shape of domains
• Nonconvex variuational problems.

Addressed to graduate and to senior undergraduate students in math and science.

Notes (preliminary):

1. Chaper 1. Introduction
2. Chapters 2 and 3 Stationarity. Development
3. Chapters 4 and 5 Inequality tests, Constrained problems. Introduction to Control theory
4. Chapter. Numerical methods. To be posted.
5. Chapters 6 and 7 Irregular solution, regularization and relaxation
6. Chapters 8 and 9 Multivariable Problems. Stationarity

Reading:

1. Robert Weinstock. Calculus of Variations with Applications to Physics and Engineering. Dover Publications, 1974.
2. I. M. Gelfand, S. V. Fomin Calculus of Variations Dover Publications, 2000

Homework 4