Entry requirements

Several Variable Calculus, or Geometry and Analysis III.

Learning outcomes

In order to pass the course the student should be able to

• give an account of the foundations of calculus of variations and of its applications in mathematics and physics;
• describe the brachistochrone problem mathematically and solve it;
• solve isoperimetric problems of standard type;
• solve simple initial and boundary value problems by using several variable calculus;
• formulate maximum principles for various equations and derive consequences;
• formulate important results and theorems covered by the course;
• use the theory, methods and techniques of the course to solve problems;
• present mathematical arguments to others.

Content

Calculus of Variations deals with optimisation problems where the variables, instead of being finite dimensional as in ordinary calculus, are functions. This course treats the foundations of calculus of variations and gives examples on some (classical and modern) applications within physics and engineering science.

The Euler–Lagrange equation. The brachistochrone problem. Minimal surfaces of revolution. The isoperimetric problem. Fermat's principle (geometric optics). Hamilton's principle (particle dynamics), Lagrange's and Hamilton's equations of motion, the Hamilton–Jacobi equation, the principle of least action. The Euler–Lagrange equation for several independent variables. Minimal surfaces. Vibrating strings and membranes, eigenfunction expansions. Quantum mechanics: the Schrödinger equation. Noether's theorem. Ritz optimisation. The maximum principle with applications.

Instruction

Lectures and problem solving sessions.

Assessment

Written and, possibly, oral examination at the end of the course. Moreover, compulsory assignments may be given during the course.