Calculus of Variations

Department:
MATH
Course Number:
7581
Hours - Lecture:
3
Hours - Lab:
0
Hours - Recitation:
0
Hours - Total Credit:
3
Typical Scheduling:
Every even spring

Minimization of functionals, Euler Lagrange equations, sufficient conditions for a minimum, geodesic, isoperimetric and time of transit problems, variational principles of mechanics, applications to control theory.

Prerequisites:

MATH 4317 or equivalent

Course Text:

No text

Topic Outline:
• The basic setup: Bernoulli and the Brachistochrone. The general setup: functionals and boundary conditions; isoperimetric problems, geodesic problems
• Minimizing in a linear space; directional derivatives; convex functions
• Convex functionals and calculus of variations; variations; sufficient conditions for minimum of convex functional -- the Euler Lagrange equation; applications in mechanics and minimum area problems
• The lemmas of DuBois-Raymond
• Minimizing without prior assumptions of smoothness, the Euler-Lagrange equations again
• Optimizing with respect to piecewise smooth functions; general linear space background, norms; the Weierstrass corner conditions
• Applications to mechanics, Lagrangians, Hamiltonians, the 2-body problem and generalizations; Hamilton-Jacobi equations
• Necessary conditions for minimization
• An introduction to control theory in the context of Calculus of Variations; examples; rocket problems