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MECH_ENG 432: The Calculus of Variations and its Applications

Quarter Offered

Fall : MWF 1:00-1:50 pm ; S. Ghosal

Prerequisites

Calculus to the level of ODEs, partial derivative and multiple integrals, some knowledge of PDEs helpful but not essential. Students should have a certain comfort level with dealing with advanced mathematical concepts.

Description

Who Takes It

The Calculus of Variations is a body of techniques for solving certain types of optimization problems. Since optimization problems are encountered in every branch of science and engineering, students with diverse backgrounds would benefit from it. 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 benefit from it if they have the prerequisites mentioned above. In the case of undergraduates, it is highly recommended that they contact the Professor prior to registering.

What It's About

This course is about a set of mathematical methods called the “Calculus of Variations” designed to solve certain kinds of optimization problems. In calculus one learns techniques to find particular values of a variable(s) for which a function is a maximum or minimum. In the Calculus of Variations, the unknown that we seek is a function or a path that maximizes or minimizes a certain quantity. It turns out that various problems in the sciences can be reformulated in the language of the Calculus of Variations and its methods then provide a unified perspective of these fields. This course stresses practical applications over mathematical rigor. After developing the foundations of the subject, we will re-examine the classical areas of physics: optics, particle mechanics, elasticity theory, vibration & waves, fluid mechanics etc. from the perspective of the Calculus of Variations.

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

Assignment/Evaluation

Homework, midterm and final exam.

Textbook

Primary source: printed lecture notes

Secondary source: a list of suggested reference texts will be provided

Contact

Professor: Sandip Ghosal

Email: s-ghosal@northwestern.edu