A question many mathematicians and physicists hear about their research is: what is this for? I believe that the meaning of this question is closely linked to the fact of what science is and how it develops. Often, a purely theoretical work is the initial step for the development of an entire area with diverse applications.

An example of this is the calculation of variations. This area of mathematics is of great importance and applicability. the use of this theory to address optimization problems, like a country's power distribution network. This, produces a limited amount of electricity that needs to be distributed to the population but on the other hand there are special services (hospitals, police, bares) which has priority if there is no energy for everyone. The problem is how to keep an electrical network running that delivers energy to the population in the most efficient way while guaranteeing these essential services. Another example is the use of this tool in the area of Biomathematics in the study of population dynamics, when the object of study is the structure of formation of complex molecules, among many other cases.

The interesting thing is to think that this area was developed from the so-called 'Brachistochrone problem'. This issue appeared in June 1669, in the German mathematics magazine Acta Eruditorium, founded by Leibniz. At that time, it was common for mathematicians to pose challenges in this way and the Swiss mathematician Johann Bernoulli proposed the following:

*Given two points A and B on a vertical plane, what is the curve drawn from one point to another such that a particle under the action of gravity alone moves in the shortest time interval?
*

Solutions to the problem were proposed by Newton, Jacob Bernoulli, Leibniz e L’Hôpital, published in the Acta Eruditorium subsequently. The curve sought is called the cycloid, name given by Galileo, who studied its properties in 1600. A cycloid is a curve defined by a point on a circle rolling without sliding along a straight line..

Bernoulli's resolution was based on Fermat's principle. It says that when light propagates between two points, she chooses the path that will have the shortest duration; e, in the refraction phenomenon, says that light propagating through different means changes its direction. Thereby, Bernoulli divided the problem into 'slices' and treated them as light passing from one medium to another. doing the calculations, he got an equation that described the cycloid curve.. The outline of the solution proposed by him can be seen below:

The method developed by Bernoulli was efficient in optimization problems and was studied and improved by Euler in 1744. The problems were getting more and more complicated to be solved with the Euler equation and, in 1762 e 1770, Lagrange published a work in which he developed an analytical method to obtain the curve in an optimization problem based on variations of functions.. And that's where the variational calculus as it is known today came about.

In summary, the brachistochrone problem is to obtain the curve that minimizes the time that a particle, under the action of gravity, it takes to get from A to B. The solution is the curve called the cycloid, which is generated by a circle rolling without sliding. Initially, the problem was treated geometrically using basic principles of optics, but it was very important for the study of optimization problems and the development of powerful tools such as variational calculus.

And this shows that science is made up of small steps and the collaboration and work of different people. The application is not immediate because what is sought is knowledge by itself. The understanding of nature and its phenomena is more than enough to answer the “what is this for?”

Reference: https://truesingularity.wordpress.com/2012/09/19/a-braquistocrona-e-o-desenvolvimento-da-ciencia/