Spring Course: MATH 479/579 Calculus of Variations
MATH 479/579 Calculus of Variations
The calculus of variations goes back to the 17th century and Isaac Newton. Newton developed the theory to solve the minimal resistance problemand later the brachistochrome problem. The theory was further elaborated by Leonhard Euler and later by Joseph-Louis Lagrange. Later mathematicians as Hamilton, Jacobi, Dirichlet and Hilbert have offered their contributions.
In 2019, Karen Uhlenbeck (University of Texas at Austin, USA), whose research is mainly in this area, received the prestigious Abel Prize. This shows that in modern times, the calculus of variations has continued to occupy center stage, witnessing major theoretical advances, along with wide-ranging applications in physics, engineering and all branches of mathematics.
The general formulation of the theory is as follows:
Let J be a functional that assigns a value to any function in some function space. Find the function that minimizes J.
Examples:
1. The three circles were released from the top point at the same time. The red one will reach the end point first.
2. A spider web consists of a lot of hanging cables.
3. A soap film connecting two parallel circles.
Minimization problems that can be analyzed by the calculus of variations serve to characterize the equilibrium configurations of almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics, string theory, and many, many others.
MATH 479/579
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