Schedule and Notes
| Week | Dates Topics | Notes | Assignments |
| 1 |
Jan. 7:
Panorama of optimization problems; scope of the course.
Jan. 9: Genesis of calculus of variations Formulation of calculus of variations problems in geometry and mechanics |
Lecture 2 Lecture 3 |
Homework #1 |
| 2 |
Jan. 14:
Happy Sankranthi! (no class)
Jan. 16: Formulation of calculus of variations problems in geometry and mechanics (contd.) A quick overview of finite variable optimization: unconstrained minimization |
|
Homework #2 |
| 3 |
Jan. 21:
Unconstrained minimization in finite number of variables: necessary and sufficient conditions Constrained minimization with equality constraints: Lagrange multiplier concept Jan. 23: Constrained minimization with inequality constraints: complementarity conditions |
|
- |
| 4 |
Jan. 28:
Karush Kuhn-Tucker (KKT) conditions Constraint qualification Sufficient condition for a constrained minimum; Bordered Hessian Jan. 30: Mathematical preliminaries for calculus of variations: functional, vector spaces, metric, normed vector space, and function spaces |
Lecture 7 Lecture 8 |
Homework #3 |
| 5 |
Feb. 4:
Gateaux variation; Frechet differential; linearity and ocntinuity of functionals. Feb. 6: Fundamental lemma of calculus of variations; Euler-Lagrange equations and their generalization to multiple derivatives and multiple functions; boundary conditions. |
Lecture 10 Lecture 11 |
Homework #4 |
| 6 |
Feb. 11:
Two and three independent variables in the integrand; boundary conditions and Euler-Lagrange equations Feb. 13: Integrand involving two independent variables and up to second derivatives; static equilibrium of a plate. |
|
Homework #5 |
| 7 |
Feb. 18:
Variational derivative;analogy to partial derivative; Euler's method of deriving the necessary conditions. Feb. 20: Global (summatve) and local (pointwise) constraints in calculus of variations; the concept of Lagrange multipliers in calculus of variations. |
Lecture 14. |
Homework #6 |
| 8 |
Feb. 25:
General (non-contemporaneous) variation; transversality conditions for functionals involving first and second derivatives Feb. 27: Weierstrass-Erdmann corner conditions; broken extremals; two-function case Intergals and invairants of Euler-Lagrange equations |
Lecture 16. |
No homework; instead go thorugh midterms of previous years |
| 9 |
Mar. 4:
Inverse problems involving Euler-Lagrange equations Variational methods in mechanics: some examples Mar. 6: Midterm examination |
Lecture 18. |
Homework #7 |
| # |
April 23:
Final examination at 3 pm to 5 pm in MMCR. |
|
Solve final exam problems of past years for practice. |
| # |
May 4:
Project presentations at 2 pm to 5 pm in MMCR. |
|
12-minute presentation per student |
You can find the content-page of the previous years here.
2013
2012
2011
2009
2007
2006
2005