Course Syllabus
Mechanical Engineering 432
The Calculus of Variations & Its Applications
Prerequisites
Calculus to the level of ODEs, partial derivative and multiple integrals, some knowledge of PDEs helpful.
Who takes it?
The calculus of variations is encountered in every branch of science and engineering, so students with diverse backgrounds would benefit from this course. In particular students with research interests in any area of mechanical engineering, physics, applied mathematics, chemistry, chemical engineering and biomedical engineering are encouraged to register. The course is primarily for graduate students (at any level) but advanced undergraduates may also register if they have the prerequisites mentioned above.
What is it about?
We often marvel at the manner in which animals seem so perfectly adapted to their environment: the cheetah is perfectly build for running at high speeds in open areas, the shark has all the adaptations for swimming at high speeds under water, certain eels use electric fields instead of vision to navigate in muddy waters and so on. Mathematically speaking each of these animals is a "solution" to an "optimization problem" that nature has solved through the iterative method of mutation and natural selection! In the machines that we build, as in natures "machines" the need to maximize or minimize certain things (such as energy consumption, cost, aerodynamic resistance etc.) is the essence of the design process. In this practically oriented course we will study "The Calculus of Variations", a powerful set of mathematical tools for solving a certain class of optimization problems that arise in diverse areas of science and engineering.
Minisyllabus
* Introduction
* Extremizing functions of several variables (review)
* Extremizing with constraints - Lagrange multipliers
* Functionals and the Euler-Lagrange equations
* Constrained optimization of functionals
* Some classical problems in the calculus of variations
* Applications: classical mechanics, geometrical optics, elasticity, fluid mechanics, vibrations and waves
Assessment/Evaluation
Home work (0 % of grade): homework will be assigned from time to time. You are encouraged to attempt them. Feel free to stop by during office hours to discuss your solution. There is nothing to turn in.
Mid Term Exam (30 % of grade): in class written exam [Fri Oct 25]
Terminal Exam (70 % of grade): in class written exam [Fri Dec 6 ]
Study Materials
Printed Lecture Notes (Primary Source): these will be posted together with a summary of the day's lecture. Please watch the "Announcements" section.
Additionally, the following books are suggested as reference material.
1. Gelfand and Fomin "Calculus of Variations" Dover
2. Feynman "The Feynman Lectures on Physics" Vol 1 Ch 26 & Vol 2 Ch 19 Addison-Wesley Longman
[Free to read online: http://www.feynmanlectures.info/ ]
3. Myskis "Advanced Mathematics for Engineers" Ch 6, Mir Publishers, Moscow
4. Courant & Hilbert "Methods of Mathematical Physics Vol 1" Ch 4, John Wiley
5. Weinstock "Calculus of Variations with applications to Physics & Engineering"
[Free pdf download https://www.pdfdrive.com/calculus-of-variations-e34313748.html
also quite inexpensive on Amazon if you prefer the book.]
Office Hours
Wednesday 3:00 - 6:00 PM Room L 495 (Tech)
Course Summary:
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