# Qns on euler-lagrangian equation

by almo
Tags: equation, eulerlagrangian
 P: 350 Do you know what a vector space is? A functional is a map whose domain is a subset of a vector space and which takes scalar values. In the context of your question a typical vector space would be the set of differentiable functions on the interval [0,1]. $V =\{ y(x)| y\, \text{is differentiable in a neighborhood of the interval}\, [0,1]\}$ An example of a functional would be a map $\mathcal{F}(y)$ with domain $\{y\in V| y(0)=1,\, y(1)=5\}$ and which is defined by a formula such as $\mathcal{F}(y) = \int_a^b \sqrt{1+(y')^2}\, dx$ In plainer language, in the context of calculus of variations, functionals take ordinary functions as inputs and return numbers as outputs. A good basic reference would be Gelfand "Calculus of Variations".