Calculus of variations

Math can help us find the best way to do things.

We can look for the shortest path between two points. A straight line is the shortest path.

Sometimes we want to find the smallest shape. You can see this with soap bubbles. They make shapes with the least area.

Light also follows a special path. It takes the shortest way through things. This helps us understand how light moves.

Many smart people studied these puzzles. Isaac Newton worked on them a long time ago. Other thinkers like Euler and Lagrange helped too.

Math helps us find these best paths.

Math can help us find the best way to do things. This field of math is called the calculus of variations. It looks for the best path or shape. We call these best paths or shapes extrema. An extremum can be a maximum or a minimum.

One puzzle is finding the shortest path between two points. On a flat surface, this is a straight line. But if the path must stay on a curved surface, it is different. We call these special paths geodesics. Light also follows a special path. It takes the path with the shortest optical length.

Another puzzle is finding a shape with the least area. You can see this with soapy water. If you dip a frame in soap, it makes a thin film. That film forms a surface with the smallest area.

Many thinkers helped build this math. Isaac Newton solved a hard problem in 1685. Later, Leonhard Euler and Joseph-Louis Lagrange added more ideas. Euler even gave the subject its name in 1756. Today, math helps us understand how things move and work.

Math can help us find the best way to do things. This field is called the calculus of variations. It looks for the best path or shape. We call these best paths or shapes extrema. An extremum can be a maximum or a minimum.

Scientists use it to solve many puzzles. One simple puzzle is finding the shortest curve between two points. On a flat surface, the answer is a straight line. However, if the path must stay on a curved surface, it is different. These special paths are called geodesics. Light also follows a special path. It takes the path with the shortest optical length through a material.

How does this math work? It uses small changes to find the best result. We call these small changes variations. We look at things called functionals. A functional is a way to turn a whole function into a single number. Often, these are written as integrals. To find the best function, mathematicians use the Euler–Lagrange equation. This equation helps us find the extremal function. An extremal is the specific function that reaches the maximum or minimum.

This process is like finding the peak of a hill. We want to find the exact spot where the change stops.

Many famous thinkers helped build this math over many years. Isaac Newton began this work in 1685. He solved a hard problem about minimal resistance. He published his work in his book, Principia, in 1687. Later, Johann Bernoulli raised a new problem in 1696. He looked at the brachistochrone curve. Newton solved this problem using variational techniques in 1697. This helped him pioneer the whole field.

Other great minds like Leonhard Euler and Joseph-Louis Lagrange added many more ideas to the theory.

History shows many important names and dates. Leonhard Euler first explained the subject in 1733. In 1756, he gave a lecture and named it the calculus of variations. Joseph-Louis Lagrange contributed greatly to the theory. Later, Adrien-Marie Legendre worked on finding maxima and minima in 1786. Many others helped, such as Carl Friedrich Gauss in 1829. Karl Weierstrass wrote a very important course on the theory. He was the first to give the math a firm foundation. In 1900, the Hilbert problems encouraged even more new development.

You can see these ideas in the real world. One example is finding a surface with the smallest area. This is called Plateau's problem. You can see this by dipping a frame in soapy water. The soap film forms a surface with the least area. This is a beautiful way to see math in action. You can also see it in how things move in physics. The principle of least action is a related idea in mechanics. Math helps us understand these shapes and movements everywhere.

The calculus of variations is a specialized branch of mathematical analysis. It focuses on finding the extrema of functionals. An extremum can be a maximum or a minimum value. While standard calculus looks for the best number in a function, this field looks for the best function in a set. This is done by studying variations, which are tiny changes made to functions.

A functional is a mapping that turns a whole function into a single real number. These mappings are often expressed as definite integrals. They involve functions and their derivatives to calculate a total value. By analyzing these variations, mathematicians can identify the specific function that produces the highest or lowest result.

To find these optimal functions, mathematicians use a specific tool called the Euler–Lagrange equation. This equation acts as a necessary condition for finding an extremum. The process begins by considering a functional and making a small change to its input function. This change is called a variation and is denoted by $\delta y$. If the functional reaches a minimum at a specific function, the variation of the functional must be zero at that point. Through the use of integration by parts, the problem transforms into a second-order ordinary differential equation. Solving this equation provides the extremal function, which is the function that achieves the extremum.

In some cases, such as when the integrand does not depend explicitly on the independent variable, the equation simplifies into the Beltrami identity.

There are different ways to classify these extrema based on the behavior of the functions. In a space of continuous functions, we distinguish between strong and weak extrema. A weak extremum is a function that minimizes or maximizes the functional within a small neighborhood where the functions and their first derivatives are close. A strong extremum is more restrictive. It requires the functional to be at an extremum even when the first derivatives are not necessarily continuous. Therefore, every strong extremum is also a weak extremum, but the reverse is not always true. Finding strong extrema is generally a much more difficult mathematical task.

The history of this field is tied to some of the greatest minds in science. Isaac Newton provided the earliest foundations in the late 17th century. In 1685, he formulated and solved the minimal resistance problem. He published this work in his famous 1687 book, *Principia*. This was the first problem in the field to be correctly solved. In 1696, Johann Bernoulli introduced the brachistochrone curve problem. Although Galileo Galilei had raised a similar problem in 1638, he did not solve it using calculus. Newton solved the brachistochrone problem in 1697 using variational techniques. This achievement helped him pioneer the entire discipline.

As the field grew, many mathematicians added layers of complexity and rigor. Leonhard Euler began to elaborate the subject in 1733. He later renamed the field the calculus of variations in his 1756 lecture, *Elementa Calculi Variationum*. This change occurred after he was influenced by the analytic approach of the 19-year-old Joseph-Louis Lagrange. Throughout the 19th century, many contributors helped refine the theory. Adrien-Marie Legendre worked on discriminating between maxima and minima in 1786. Carl Friedrich Gauss contributed in 1829, and Carl Jacobi worked on it in 1837. Later, Karl Weierstrass provided an epoch-making course that placed the theory on a firm, unquestionable foundation.

We can see the calculus of variations working in many physical phenomena. One classic example is finding the shortest path between two points. On a flat plane, the solution is a straight line. However, if the path must stay on a curved surface, the solution is called a geodesic. Another example is Fermat's principle, which states that light follows the path of shortest optical length. This path depends on the material of the medium through which the light travels. In mechanics, the principle of least or stationary action serves as a related concept.

Another fascinating application involves finding surfaces of minimal area. This is known as Plateau's problem, which asks for a surface that spans a specific contour. You can observe this physically by dipping a wire frame into soapy water. The soap film will naturally form a surface with the smallest possible area. While these experiments are easy to perform, the math behind them is quite complex. Such surfaces can have non-trivial topology or multiple locally minimizing solutions. This field continues to evolve through modern developments like optimal control theory and dynamic programming.