# Calculus of Variations: Course Outline (MAT 466)

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers.

## Course Objective

At the end of this course the students will be able to understand the calculus of variation with the problem of extrmizing “functionals”. One of the objectives of a course is to prepare students for future courses in their areas of specialization for solving practical problems.

## Course Contents

Basics of functional variations, General statement of the external problems, Maxima and minima, weak local minima and maxima, well-posed end point conditions, Existence and uniqueness of solutions, Simple Eulerian maximization problems,  Euler-Lagrange conditions, First integrals of the Euler-Lagrange equations, Canonical formalism of the Euler-Lagrange conditions, Action integrals and their functional variations, Hamilton conditions, Inverse problems in variational calculus, Isoperimetric problems, Constrained surfaces of least curvature, Broken externals, Weierstrass-Erdmann conditions, Multidimensional cases and higher order  necessary conditions, Lagrange problem and the Euler-Lagrange theorem in multidimensions, Applications to the Branchistochrone, Minimum surface of revolution, Geodesics, Geometrical optics, Fermat principle, Hamilton equations of motion, Eigenvalue and eigenfunction problems, Ritz Variational principle, Strum-Liouville problems, Membrane vibrations, Schrodinger equation and energy minimization, Existence of minima of Dirichlet integral.