least surface of revolution
The points and have to be by an arc such that when it rotates around the -axis, the area of the surface of revolution (http://planetmath.org/SurfaceOfRevolution) formed by it is as small as possible.
The area in question, expressed by the path integral
along , is to be minimised; i.e. we must minimise
Since the integrand in (2) does not explicitly depend on , the Euler–Lagrange differential equation (http://planetmath.org/EulerLagrangeDifferentialEquation) of the problem, the necessary condition for (2) to give an extremal , reduces to the Beltrami identity
by integration of which we choose the new constant of integration such that when :
We can write two equivalent (http://planetmath.org/Equivalent3) results
Adding these yields
From this we see that the extremals of the problem are catenaries. It means that the least surface of revolution in the question is a catenoid.
|Title||least surface of revolution|
|Date of creation||2013-03-22 19:12:11|
|Last modified on||2013-03-22 19:12:11|
|Last modified by||pahio (2872)|