MA431 ‐ Calculus of Variations (Spring 2024)

Section

MA431-01 — MTRF 1:00-1:50

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Prerequisite: MA330 - Vector Calculus

Textbook: The Calculus of Variations by Bruce van Brunt. It is available to download for free on SpringerLink through Logan Library.

A familiar and important application of calculus is to optimize a function, i.e. to find a number \(x\) that produces a maximum or minimum value of \(f(x)\). In MA431 - Calculus of Variations (Course Catalog) we will consider optimization questions like the following, which are qualitatively different than those encountered in basic calculus:

What is the shape of the shortest curve between two points on a surface? For example, you "know" that the shortest path between two points in a plane is a straight line, but how would you prove it rigorously?
For a given perimeter length, what curve encloses the largest possible area? Or for a given surface area, what surface encloses the largest possible volume?
What is the free motion of a particle from point A to point B? The trajectory curve \(\vec{c}(t)\) that the particle follows is the one that minimizes action, which involves a difference of the particle's kinetic and potential energies along that trajectory: \[ \text{Action} = \int_{t_1}^{t_2} \left[ \text{KE}(\vec{c}(t)) - \text{PE}(\vec{c}(t)) \right] dt. \]
What is the shape of the surface that a soap film will form between two circular rings that are sufficiently close together?

Source: Collapse of soap bubble catenoid in slow motion

To answer these questions, the goal is to find an unknown function that maximizes or minimizes a functional, which is a rule that takes a function as its input and returns a specific numerical value as its output. For example, the arc length of a curve \(y=f(x)\) on \(a \leq x \leq b\) defines a functional: \[ \underbrace{L(f)}_{\text{Input: a function} \ f \ \text{on} \ [a,b]} = \quad \underbrace{\int_a^b \sqrt{1 + [f'(x)]^2} \, dx}_{\text{Output: a number}}. \]

The Calculus of Variations is about finding critical points of functionals, and the theory/computations will build upon ideas from single/multivariable calculus and differential equations!

The use of the Calculus of Variations in solving partial differential equations (PDE) will also be discussed as interest and time permits.