# What does “Calculus of variations” mean?

The field of mathematics known as the “Calculus of variations” is a branch of calculus that deals with studying and optimizing functionals. Functionals are mathematical expressions that involve functions instead of just variables. In essence, the Calculus of variations seeks to find the path or function that minimizes or maximizes a given functional.

## Examples of usage

To better understand the concept of the Calculus of variations, let’s explore a few examples where it finds practical applications:

### 1. Brachistochrone Problem

One classic problem tackled by the Calculus of variations is the Brachistochrone problem. It involves finding the curve between two points in which a bead, subject only to the force of gravity, will slide from one point to the other in the shortest amount of time.

To solve this problem, we can set up a functional representing the time taken for the bead to travel along a certain curve. By applying the Euler-Lagrange equation, which is a fundamental tool in the Calculus of variations, we can obtain the curve that minimizes the functional and represents the fastest path between the two points.

### 2. Geodesics on Surfaces

Geodesics are the shortest paths between two points on a surface. The Calculus of variations comes into play when determining these geodesics on curved surfaces. By considering the surface as a functional and applying the Euler-Lagrange equation, we can find the geodesics that minimize the functional and represent the shortest paths.

For example, on the Earth’s surface, the Calculus of variations helps in determining the shortest distance between two cities, taking into account the curvature of the Earth.

### 3. Minimal Surface Problems

The study of minimal surfaces, which are surfaces with the least possible area, involves the Calculus of variations. For instance, soap films naturally form minimal surfaces due to the surface tension minimizing their area.

By treating the minimal surface as a functional and applying the Euler-Lagrange equation, mathematicians can find the surface that minimizes the area under given constraints. This has implications in various fields such as architecture, physics, and materials science.

### 4. Fermat’s Principle of Least Time

Fermat’s principle states that light takes the path that requires the least time when traveling between two points. The Calculus of variations enables us to prove this principle mathematically by treating the path of light as a functional.

By considering all possible paths taken by light and minimizing the corresponding functional, we can derive the laws of reflection and refraction, providing a solid foundation for the study of optics.

## Conclusion

In conclusion, the “Calculus of variations” is a mathematical field concerned with optimizing functionals. Through the application of the Euler-Lagrange equation, this branch of calculus allows us to determine the path or function that minimizes or maximizes a given functional. The examples mentioned above provide insights into the practical applications of the Calculus of variations in various domains such as mechanics, geometry, and optics.