Introduction to the Calculus of Variations

Course Aim

Provide an introduction to the foundations and applications of the calculus of variations.

Course Description

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.

Course Contents

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%

Prerequisites or Prior Knowledge

Students should have a robust understanding of undergraduate-level calculus, single and multivariable. Familiarity with differential equations would also be beneficial.


Mark Got. A First Course in the Calculus of Variations. American Mathematical Society. Providence, 2014.

Reference Books

Peter J. Olver. The Calculus of Variations. Download from:∼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

Research Specialties