Calculus of variations

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.