The Calculus of Variations course is composed of five parts:
— Modeling of Some Problems in Calculus of Variations. In this section, we begin by introducing the fundamental concepts of calculus of variations. To do so, we examine various shortest path problems on surfaces, as well as shape optimization problems (such as the Brachistochrone problem, the fastest tunnel, soap bubbles, Huygens’ isochronous pendulum, and isoperimetric problems like the catenary and Dido’s problem). We emphasize understanding these concepts and their application to physical modeling.
— Optimization Problems in Functional Spaces. We address optimization problems in function spaces. We also review the theory of differential calculus in Banach spaces and analyze the properties of the dual space.
— Major Theorems of Calculus of Variations. This section presents proofs of the major theorems in calculus of variations, including those of Euler, Erdmann, Legendre, and Weierstrass. We discuss their theoretical significance and practical implications for problem-solving.
— Analysis of the Existence of Solutions. We study in detail the conditions for the existence of solutions to problems in calculus of variations. This includes introducing advanced concepts such as compactness in functional spaces and weak convergence. In this section, we also demonstrate important results from functional analysis (Ascoli-Arzelà, Hahn-Banach separation theorem).
— Practical Applications and Numerical Methods. Finally, we develop numerical methods for solving certain problems in calculus of variations. Students will have the opportunity to implement these methods through practical exercises (in Julia).
Learning Objectives:
- Analyze optimization problems in functional spaces
- Model problems in calculus of variations
- Apply the fundamental theorems of Euler-Lagrange and Weierstrass
- Analyze the existence and nature of extremals
- Master the concepts of compactness and weak convergence in reflexive spaces
- Know how to discretize a calculus of variations problem and apply optimization methods (Gradient, Newton, Quasi-Newton), as well as available software in Julia (Ipopt, Snopt)
- Enseignant: Zidani Housnaa