Calculus of Variations

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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.


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