The Math of Bubbles // Minimal Surfaces & the Calculus of Variations #SoME3

The video begins by exploring the concept of minimal surfaces, such as the catanoid formed when dipping two rings into a bubble solution. The mathematical challenge is to determine the shape of a bubble given a specific boundary, which is the essence of the study of minimal surfaces. The presenter demonstrates the use of a helix to create a helicoid shape and introduces the calculus of variations as a technique to derive these surfaces. The focus is on finding the surface with the least area between two parallel rings, which is simplified by assuming circular symmetry. The problem is reformulated to find the curve or function that, when rotated around the x-axis, yields the smallest surface area. A formula for computing the surface area of a function revolved around the x-axis is provided, and the concept of functionals is introduced as a function that takes other functions as input.

The video continues by delving into the calculus of variations, a mathematical framework for finding functions that minimize a certain integral. The presenter illustrates this with the example of a function f(x) that minimizes a given integral expression. To find such a function, the video describes a method of perturbing f(x) and considering the resulting changes. A new function G(x) is defined as a sum of the original function f(x) and a perturbation function P(x), scaled by a small factor epsilon. The integral expression is then evaluated for G(x), resulting in a function Phi of epsilon. The minimization property is applied, setting the derivative of Phi with respect to epsilon to zero. This leads to the use of the multivariable chain rule to find the derivative of the Lagrangian, a function that depends on the independent variable x, the function f(x), and its derivative. The video concludes this section by emphasizing the generality of the approach, noting that many interesting Lagrangians come from physics.

The presenter then focuses on solving the Euler-Lagrange equation, which is derived from the calculus of variations and is key to finding the function that minimizes the integral. The specific Lagrangian from the surface area formula is used, and the partial derivatives with respect to the function and its derivative are computed. The process simplifies the equation and leads to an integral that must equal a constant. The video demonstrates how to isolate the derivative of the function, integrate both sides, and recognize the resulting expression as the inverse of hyperbolic cosines. The constants from the integration are discussed, and the function f(x) is expressed in terms of these constants.

The video concludes with the application of the derived function to match a specific pair of circles, which represent the boundary of the bubble. The constants in the hyperbolic cosine expression are adjusted to fit the boundary conditions. The presenter uses numerical approximation to demonstrate how the function can be made to fit the given circles. However, it is noted that if the circles are too small or too far apart, it may not be possible to find a matching bubble shape, leading to a physical limitation where the bubble would 'pop' and result in two flat discs. The video wraps up by stating that the catanoid, formed by revolving the hyperbolic cosine function around the x-axis, represents the minimal surface area for the given boundary. The presenter acknowledges the vastness of the topic of minimal surfaces and encourages viewers to explore further.