Calculus of Variations M-575

Instructor Andrej Cherkaev

Course content

The main question of the classical theory of Calculus of Variation is: What curve or surface is in a sense `the best one'? For example,

• What curve with the fixed length covers maximal area? (of course, the circle!)
• What curve corresponds to the fastest glide along it? (the brachistochrone problem)
• What surface with a fixed boundary has the smallest area? (the minimal surface problem)
• What variable thickness of a plate maximizes its stiffness? (an optimal design problem).

In these problems, the extremal property is attributed to an entire curve (function). A group of methods aimed to find `optimal' functions is called Calculus of Variations. The Calculus of Variations has been originated by Bernoulli, Newton, Euler; systematically developed beginning from XVIII century; it still attracts attention of mathematicians and it helps scientists and engineers to find the best possible solutions. The importance of Calculus of Variations is emphasized by extremal principles in nature: in mechanics, a real motion minimizes the energy; in thermodynamics, the dissipative processes maximizes the dissipation rate, etc. This is why Calculus of Variations forms a basis of Physics, Mechanics, Numerical Methods, etc. The modern branches of Calculus of Variations include Optimization Theory, Optimal Control, Structural Optimization, etc.

The course covers classical techniques of Calculus of Variations, discusses natural variational principles in classical and continuum mechanics, and introduces modern applications: control theory, numerical methods, etc. Clear and elegant methods of Calculus of Variations allow us to solve a lot of problems in Physics and Engineering which will be in the focus of the course.

Text

1. Robert Weinstock. Calculus of Variations with Applications to Physics and Engineering. Dover 1974.
2. Notes.

Prerequisites

M 251, M 252, M 353 or equivalents.