Introduction to the Calculus of Variations
Provide an introduction to the foundations and applications of the calculus of variations.
Students successfully completing this course will:
• Understand the basic concepts of the calculus of variations, including the notion of a functional, admissible classes of functions and variations, the first and second variation conditions, and the fundamental localization lemma.
• Gain proficiency in applying the first and second variation conditions and in deriving Euler–Lagrange equations.
• Distinguish between essential and natural boundary conditions, and understand their significance in variational calculus.
• Understand variable endpoint conditions and the Weierstrass–Erdmann corner conditions. • Acquire the ability to solve simple boundary-value problems stemming from variational principles with one or multiple independent variables.
• Explore global and local constraints and learn how to apply those principles to formulate and solve constrained variational problems.
• Develop skills in formulating and applying the calculus of variations to solve practical problems in physics and engineering.
• Learn to analyze and interpret the solutions obtained from solving variational problems.
• Learn to apply the Ritz and Galerkin methods.
The calculus of variations originated from classical investigations into fundamental problems of maximizing enclosed areas, minimizing travel times, determining geodesics, and optimizing trajectories in mechanics. Variational problems involve the optimization of functionals, which are real-valued objects which take functions as inputs. This course will offer a comprehensive exploration of the conceptual basis and methods of the calculus of variations, including functionals, necessary conditions for optimality, necessary and sufficient conditions for optimality, essential and natural boundary conditions, variable end-point conditions, the treatment of global and local constraints, and direct methods. Applications will span geometry, physics, and engineering, revealing the pivotal role of variational methods in describing natural phenomena and in optimizing processes and systems. Through practical problem-solving exercises, students will become proficient in formulating extracting information from and variational principles. The course will provide a foundation for further exploration in various fields, including advanced physics, engineering applications, or interdisciplinary studies where variational methods find widespread application.
1. Review of classical optimization
2. Functionals
3. Admissible sets of competitors and variations
4. First and second variation conditions
5. Localization, Euler–Lagrange equations
6. Essential and natural boundary conditions
7. Conversion of the second variation condition to an eigenvalue problem
8. Variable endpoint conditions
9. Weierstrass–Erdmann corner conditions
10. Sufficient conditions for optimality
11. Ritz and Galerkin methods
Homework: 40%
Project: 60%
Students need:
• a keen interest in mathematical abstraction and its practical applications;
• a robust understanding of undergraduate-level single and multivariable calculus, linear algebra, and some exposure to differential equations;
• a curiosity for optimization problems and a willingness to engage in theoretical reasoning and problem-solving;
Mark Got. A First Course in the Calculus of Variations. American Mathematical Society. Providence, 2014.
Peter J. Olver. The Calculus of Variations. Download from: http://www.math.umn.edu/∼olver
Stefan Hildebrandt & Anthony Tromba. The Parsimonious Universe. Copernicus. Springer-Verlag. New York, 1996.
Don S. Lemons. Perfect Form. Princeton University Press. Princeton, 1997.
Alternate years course, even years. Alternates with A104