Homework #4
Assigned: Feb. 7th, 2012
Due: Feb. 14, 2012
Points: 50
Additional points for work that is beyond instructor's expectation!

1. 25 points
   Write down the Euler-Lagrange equations and boundary conditions for unconstrained minimization of a functional of n functions of three independent variables and their m derivatives in as general terms as possible.
2. 25 points
   Prepare a PowerPoint (only .ppt, i.e., compatibility mode 1998-2007 version) presentation with not more than 8 slides for one of the following problems in the format prescribed ahead. Each student was earmarked for one of these problems in the class. Please out down your name and gmail address because I will use these, after some editing, in my NPTEL course on this subject.

   Format
   Slide 1: Statement of the problem with suitable figures and text. Use another slide (1a) if you need.
   Slide 2: Necessary conditions and differential equation and boundary conditions.
   Slides 3 to 8: Solution, method of solution, plot of the solution, physical interpretation, and any nuances, extensions, and subtleties.

   List of problems
   

   1. Straightline problem
   2. Brachistochrone problem
   3. Hanging chain problem
   4. Catenoid and minimum surface of revolution problem
   5. Geodesic on a single-sheet hyperboloid
   6. Maximum enclosed area for a given length of a curve in a plane
   7. Minimal surface with a given 3D curve as its boundary (soap film problem)
   8. Boat traversal problem
   9. Circling airplane problem
   10. Chatterjee's problem
   11. Minimum potential energy problem for static deformation of a beam
   12. Hamilton's principle for a spring-mass system
   13. Minimal distance path from one point to another with an obstacle in between
   14. Ajay Dangi's choice problem
   15. Heat conduction problem