classical isoperimetric problem
The points and on the -axis have to be by an arc with a given length (http://planetmath.org/ArcLength) such that the area between the -axis and the arc is as great as possible.
Denote the equation of the searched arc by . The task, which belongs to the isoperimetric problems (http://planetmath.org/IsoperimetricProblem), can be formulated as
under the constraint condition
We have the integrands
The variation problem for the functional in (1) may be considered as a free variation problem (without conditions) for the functional where is a Lagrange multiplier. For this end we need the Euler–Lagrange differential equation (http://planetmath.org/EulerLagrangeDifferentialEquation)
Since the expression does not depend explicitly on , the differential equation (3) has, by the Beltrami identity, a first integral of the form
which reads simply
This differential equation may be written
where one can separate the variables (http://planetmath.org/SeparationOfVariables) and integrate, obtaining the equation
of a circle. Here, the parametres may be determined from the conditions
Thus the extremal of this variational problem is a circular arc (http://planetmath.org/CircularSegment) 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|
|Date of creation||2013-03-22 19:10:26|
|Last modified on||2013-03-22 19:10:26|
|Last modified by||pahio (2872)|