Variational Calculus for Optimization

This crash course provides an introduction to the Calculus of Variations, a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers.

Variational principles are fundamental in many areas of science and engineering, most notably in classical mechanics (Principle of Least Action), optics (Fermat’s Principle), and differential geometry (geodesics). They also provide foundational concepts for understanding advanced optimization techniques, duality, and even some aspects of machine learning, such as regularization and optimal control.

What You’ll Learn:

Over five parts, this series will guide you through:

1. Functionals and the First Variation: Understanding what functionals are and how to define a “derivative” for them to find optimal functions.
2. The Euler-Lagrange Equation: Deriving the central differential equation that functions extremizing a functional must satisfy.
3. Lagrangian, Hamiltonian, and the Legendre Transform: Exploring the deep connections between variational calculus and classical mechanics, and introducing the Legendre transform as a bridge to duality.
4. Classic Examples and Special Cases: Applying the Euler-Lagrange equation to solve famous problems like the shortest path, the brachistochrone, and the catenary, and discussing useful simplifications.
5. Generalizations and Constraints: Extending variational methods to functionals involving higher-order derivatives, multiple functions, multiple independent variables, and handling constrained variational problems.

By the end of this crash course, you’ll have a solid understanding of the core principles of variational calculus and how they are used to solve problems involving the optimization of functions. This will lay the groundwork for further study in advanced optimization and related fields.