Home > 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.
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 surfaceIn mathematics, a minimal surface is a surface with mean curvature of zero, or, equivalently, a surface of minimum area subject to constraints on the location of its boundary. Examples of minimal surfaces include catenoids and helicoids. A soap film stret ( soap bubbleA soap bubble is a very thin film of soap water that forms a hollow spherical shape with an iridescent surface. Soap bubbles usually last for only a few moments and burst either on their own or on contact with another object. Due to their fragile nature t) problem, known as Plateau's problemPlateau's problem is to find the minimal surface with a given boundary. It is named after Joseph Plateau, who was interested in soap films, but was raised by Joseph Louis Lagrange in 1760. A general solution was first given in 1931 by Jesse Douglas, who w.
See also
- Isoperimetric inequality
- Functional analysisFunctional analysis Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. It has its historical roots in the study of transformations such as the Fourier transform and in t
- Pekeris and the HeliumHelium is a colorless, odorless, tasteless chemical element, one of the noble gases of the periodic table of elements. Its boiling and melting points are the lowest among the elements; except in extreme conditions, it exists only as a gas. The second most atomFor alternative meanings see atom (disambiguation). An atom is a microscopic structure found in all ordinary matter around us. Atoms are composed of 3 types of subatomic particles: electrons, which have a negative charge; protons, which have a positive ch
- Variational principle
- Fermat's principle
- principle of least action
External links
Mathematical analysis
