Calculus of variations

Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. Such 'functionals' can for example be formed as integrals involving an unknown function and its derivatives. The interest is in extremal functions: those making the functional attain a maximum or minimum value. Some classical problems on curves were posed in this form: one example is the brachistochrone, the path along which a particle would descend under gravity in the shortest time from a given point A to a point B not directly beneath it. Amongst the curves from A to B one has to minimise the expression representing the time of descent.

The key theorem of calculus of variations is the Euler-Lagrange equation. This corresponds to the stationary condition on a functional. As in the case of finding the maxima and minima of a function, the analysis of small changes round a supposed solution gives a condition, to first order. It cannot tell one directly whether a maximum or minimum has been found.

Variational methods are important in theoretical physics: in Lagrangian mechanics and in application of the principle of stationary action to quantum mechanics. They were also much used in the past in pure mathematics, for example the use of the Dirichlet principle for harmonic functions by Bernhard Riemann.

In modern mathematics the calculus of variations as such is no longer much used. The same material can appear under other headings, such as Hilbert space techniques, Morse theory, or symplectic geometry. The term variational is used of all extremal functional questions. The study of geodesics in differential geometry is a field with an obvious variational content. Much work has been done on the minimal surface (soap bubble) problem, known as Plateau's problem.