Calculus of variations

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, which depends upon the material of the medium. One corresponding concept in mechanics is the principle of least/stationary action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet's principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in soapy water. Although such experiments are relatively easy to perform, their mathematical formulation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

The calculus of variations began with the work of Isaac Newton, such as with Newton's minimal resistance problem, which he formulated and solved in 1685, and later published in his Principia in 1687, which was the first problem in the field to be formulated and correctly solved, and was also one of the most difficult problems tackled by variational methods prior to the twentieth century. This problem was followed by the brachistochrone curve problem raised by Johann Bernoulli (1696), which was similar to one raised by Galileo Galilei in 1638, but he did not solve the problem explicitly nor did he use the methods based on calculus. Bernoulli solved the problem using the principle of least time in the process, but not calculus of variations. In 1697 Newton solved the problem using variational techniques, and as a result, he pioneered the field with his work on the two problems. The problem would immediately occupy the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733. Joseph-Louis Lagrange was influenced by Euler's work to contribute greatly to the theory. After Euler saw the 1755 work of the 19-year-old Lagrange, Euler dropped his own partly geometric approach in favor of Lagrange's purely analytic approach and renamed the subject the calculus of variations in his 1756 lecture Elementa Calculi Variationum.

Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. Isaac Newton and Gottfried Leibniz also gave some early attention to the subject. To this discrimination Vincenzo Brunacci (1810), Carl Friedrich Gauss (1829), Siméon Poisson (1831), Mikhail Ostrogradsky (1834), and Carl Jacobi (1837) have been among the contributors. An important general work is that of Pierre Frédéric Sarrus (1842) which was condensed and improved by Augustin-Louis Cauchy (1844). Other valuable treatises and memoirs have been written by Strauch^[which?] (1849), John Hewitt Jellett (1850), Otto Hesse (1857), Alfred Clebsch (1858), and Lewis Buffett Carll (1885), but perhaps the most important work of the century is that of Karl Weierstrass. His celebrated course on the theory is epoch-making, and it may be asserted that he was the first to place it on a firm and unquestionable foundation. The 20th and the 23rd Hilbert problem published in 1900 encouraged further development.

In the 20th century David Hilbert, Oskar Bolza, Gilbert Ames Bliss, Emmy Noether, Leonida Tonelli, Henri Lebesgue and Jacques Hadamard among others made significant contributions. Marston Morse applied calculus of variations in what is now called Morse theory. Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory. The dynamic programming of Richard Bellman is an alternative to the calculus of variations.

The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements ${\displaystyle y}$ of a given function space defined over a given domain. A functional ${\displaystyle J[y]}$ is said to have an extremum at the function ${\displaystyle f}$ if ${\displaystyle \Delta J=J[y]-J[f]}$ has the same sign for all ${\displaystyle y}$ in an arbitrarily small neighborhood of ${\displaystyle f.}$ The function ${\displaystyle f}$ is called an extremal function or extremal. The extremum ${\displaystyle J[f]}$ is called a local maximum if ${\displaystyle \Delta J\leq 0}$ everywhere in an arbitrarily small neighborhood of ${\displaystyle f,}$ and a local minimum if ${\displaystyle \Delta J\geq 0}$ there. For a function space of continuous functions, extrema of corresponding functionals are called strong extrema or weak extrema, depending on whether the first derivatives of the continuous functions are respectively all continuous or not.

Both strong and weak extrema of functionals are for a space of continuous functions but strong extrema have the additional requirement that the first derivatives of the functions in the space be continuous. Thus a strong extremum is also a weak extremum, but the converse may not hold. Finding strong extrema is more difficult than finding weak extrema. An example of a necessary condition that is used for finding weak extrema is the Euler–Lagrange equation.