# Simplest Variational Problem

## Problem

Let $ F\paren{x,y,z}$ be a function of a differentiability class $C^2$ WRT all its arguments.

Let $y:\R\to\R$ be a continuously differentiable function for $x\in\sqbrk{a,b}$ such that

- $y\paren a=A$

- $y\paren b=B$

Then among all functions $y$ find the one for which the functional

- $\displaystyle J\sqbrk y=\int_a^b F\paren{x,y,y'}\rd x$

has a weak extremum.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 1.4$: The Simplest Variational Problem. Euler's Equation