classical isoperimetric problem

The points a and b on the x-axis have to be by an arc with a given length ( l such that the area between the x-axis and the arc is as great as possible.

Denote the equation of the searched arc by  y=y(x).  The task, which belongs to the isoperimetric problemsMathworldPlanetmath (, can be formulated as

to maximiseaby𝑑x (1)

under the constraint condition

ab1+y 2𝑑x=l. (2)

We have the integrands

f(x,y,y)y,g(x,y,y)1+y 2.

The variation problem for the functional in (1) may be considered as a free variation problem (without conditions) for the functional ab(f-λg)𝑑x where λ is a Lagrange multiplier.  For this end we need the Euler–Lagrange differential equationMathworldPlanetmath (

y(f-λg)-ddxy(f-λg)= 0. (3)

Since the expression f-λg does not depend explicitly on x, the differential equation (3) has, by the Beltrami identityMathworldPlanetmath, a first integral of the form


which reads simply

y-λ1+y 2=C2.

This differential equation may be written


where one can separate the variables ( and integrate, obtaining the equation


of a circle.  Here, the parametres C1,C2,λ may be determined from the conditions

y(a)=y(b)= 0,arc length=l.

Thus the extremal of this variational problem is a circular arc ( connecting the given points.

Note that in every point of the arc, the angle of view of the line segment between the given points is constant.

Title classical isoperimetric problem
Canonical name ClassicalIsoperimetricProblem
Date of creation 2013-03-22 19:10:26
Last modified on 2013-03-22 19:10:26
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 22
Author pahio (2872)
Entry type Example
Classification msc 47A60
Classification msc 49K22
Classification msc 49K05
Related topic LagrangeMultiplierMethod
Related topic CircularSegment
Related topic AngleOfViewOfALineSegment