Mechanical Engineering, Indian Institute of
Science, Bangalore 560 012, India
Optimization hinders evolution!
ME256
Variational methods and structural optimization
Aug.-Dec. 2005
Instructor: G. K.
Ananthasuresh, Room 106, ME Building, suresh at
mecheng.iisc.ernet.in
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Homework #1
Assigned: Aug. 4th, 2005
Due: Aug. 11th, 2005
Points: 20
-
10 points
Understanding what a functional is crucial to the study of calculus of
variations. We can reinforce our understanding by formulating functionals
pertaining to practical problems. Here are three simple examples. Derive
the
functional in each case.
- Given two points in a plane, (x1,y1) and
(x2,y2), find a curve y(x)
in the domain [x1,x2] such its surface of revolution
is minimum.Write the functional for this problem and also note the
difference between this and the next problem.
- Find a curve y(x) of length L defined on x = [a,b] so that the
surface
of
revolution of y(x) about the x-axis is minimized. Derive the functionals
(objective and constraint expressions)
using which this problem can be solved using calculus of variations.
-
A wire is bent into the shape of a closed curve in three dimensions. Its
shape is known to us. Assume the shape of the curve in parametric form.
Now, find the surface of
minimum area which is bounded by
the bent wire. Derive expressions for the functionals describing the
objective and constraint in this problem.
-
10 points
Thanks to many mathematicians of the past, we now have the tools of
calculus of variations that help us optimize functionals in a systematic
manner. The solution method that we know today did not come about
overnight. Many great individuals who laid the foundation for this field
used geometric and ordinary calculus methods
to find the optimizing functions for a given functional. Each of you will
be given a (different) handout of one of the solutions given by Fermat,
Galileo, Newton, Bernoullis, Leinitz, Euler, Lagrange, etc., for some classical
problems of calculus of variations. Your task is to go through it and
ponder over how they did it WITHOUT using the systematic method we use
today. You then have to write your comments on it.
Here are the options:
- Fermat's solution for Snell's law--necessary condition
- Fermat's solution for Snell's law--sufficiency proof (already taken by Girish)
- Galileo's (incorrect) solution to the brachistochrone problem (BP)
- John Bernoulli's solution to BP
- Leibnitz's solution to BP
- Newton's solution to BP (already taken by Saurav)
- James Bernoulli's solution to BP
- Euler's general solution to calculus of variations problems
- Lagrange's derivation of necessary conditions to calculus of variations problems