A simple example of such a problem is to find the curve of shortest length connecting two points.
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.
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.
[2] It immediately occupied the attention of Jacob Bernoulli and the Marquis de l'Hôpital, but Leonhard Euler first elaborated the subject, beginning in 1733.
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.
[3][4][b] Adrien-Marie Legendre (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima.
An important general work is that of Pierre Frédéric Sarrus (1842) which was condensed and improved by Augustin-Louis Cauchy (1844).
(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.
[5] 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.
[6] Lev Pontryagin, Ralph Rockafellar and F. H. Clarke developed new mathematical tools for the calculus of variations in optimal control theory.
[7][8][9][c] The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals.
The maxima and minima of a function may be located by finding the points where its derivative vanishes (i.e., is equal to zero).
In general this gives a second-order ordinary differential equation which can be solved to obtain the extremal function
In this case, this quantity is the Hamiltonian, the Legendre transform of the Lagrangian, which (often) coincides with the energy of the system.
The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral
Within a convex area and a positive thrice differentiable Lagrangian the solutions are composed of a countable collection of sections that either go along the boundary or satisfy the Euler–Lagrange equations in the interior.
However Lavrentiev in 1926 showed that there are circumstances where there is no optimum solution but one can be approached arbitrarily closely by increasing numbers of sections.
There are several results that gives criteria under which the phenomenon does not occur - for instance 'standard growth', a Lagrangian with no dependence on the second variable, or an approximating sequence satisfying Cesari's Condition (D) - but results are often particular, and applicable to a small class of functionals.
Plateau's problem consists of finding a function that minimizes the surface area while assuming prescribed values on the boundary of
[k] Eventually it was shown that Dirichlet's principle is valid, but it requires a sophisticated application of the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998).
If these forces are in equilibrium, then the variational problem has a solution, but it is not unique, since an arbitrary constant may be added.
This result depends upon the regularity theory for elliptic partial differential equations; see Jost and Li–Jost (1998) for details.
Fermat's principle states that light takes a path that (locally) minimizes the optical length between its endpoints.
As this calculation demonstrates, Snell's law is equivalent to vanishing of the first variation of the optical path length.
In order to find such a function, we turn to the wave equation, which governs the propagation of light.
Wave fronts for light are characteristic surfaces for this partial differential equation: they satisfy
Hence, solving the associated partial differential equation of first order is equivalent to finding families of solutions of the variational problem.
Analogy with Fermat's principle suggests that solutions of Lagrange's equations (the particle trajectories) may be described in terms of level surfaces of some function of