A central result in the calculus of variations which addresses problems such as finding curves of minimal length (geodesics) or solving the brachistochrone problem. A functional I associates a real number

(where a, b are fixed limits) with each of a family of graphs y(x); this functional might represent the arc length of such graphs, the time taken to travel along the graphs under gravity, etc. Euler’s equation then states I is maximal when

As an example, the arc length of a curve y(x) from (a,c) to (b,d) is

so that $F\left(x,y,y^{\prime }\right)=\sqrt{1+{\mathrm{y\prime }}^2}$ and Euler’s equation then reads

Integrating and rearranging we find that y′ is constant, and so the graph of y is a straight line.