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.
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.
