In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian-French mathematician Joseph-Louis Lagrange in the 1750s. Because a differentiable functional is stationary at its local maxima and minima, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional...

In mathematics, the term functional (as a noun) has at least two meanings. In modern linear algebra, it refers to a linear mapping from a vector space V {\displaystyle V} into its field of scalars, i.e., to an element of the dual space V ∗ {\displaystyle V^{*}} . In mathematical analysis, more generally and historically, it refers to a mapping from a space X {\displaystyle X} into the real numbers, or sometimes into...