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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