least surface of revolution

The points  $P_1=(x_1,y_1)$  and  $P_2=(x_2,y_2)$  have to be by an arc $c$ such that when it rotates around the $x$-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

$A=\mathrm{\hspace{0.33em}2}\pi {\displaystyle {\int }_{P_1}^{P_2}}y𝑑s,$ (1)

along $c$, is to be minimised; i.e. we must minimise

${\displaystyle {\int }_{P_1}^{P_2}}y𝑑s={\displaystyle {\int }_{x_1}^{x_2}}\sqrt{1+y^{\prime 2}}|dx|.$ (2)

Since the integrand in (2) does not explicitly depend on $x$, the Euler–Lagrange differential equation (http://planetmath.org/EulerLagrangeDifferentialEquation) of the problem, the necessary condition for (2) to give an extremal $c$, reduces to the Beltrami identity

$$y\sqrt{1+y^{\prime 2}}-y^{\prime }\cdot \frac{y{y}^{\prime }}{\sqrt{1+y^{\prime 2}}}\equiv \frac{y}{\sqrt{1+y^{\prime 2}}}=a,$$

where $a$ is a constant of integration.  After solving this equation for the derivative $y^{\prime }$ and separation of variables, we get

$$\pm \frac{dy}{\sqrt{y^2-a^2}}=\frac{dx}{a},$$

by integration of which we choose the new constant of integration $b$ such that  $x=b$  when  $y=a$:

$$\pm {\int }_a^y\frac{dy}{\sqrt{y^2-a^2}}={\int }_b^x\frac{dx}{a}$$

We can write two equivalent (http://planetmath.org/Equivalent3) results

$$\ln \frac{y+\sqrt{y^2-a^2}}{a}=+\frac{x-b}{a},\ln \frac{y-\sqrt{{}^2-a^2}}{a}=-\frac{x-b}{a},$$

i.e.

$$\frac{y+\sqrt{y^2-a^2}}{a}=e^{+\frac{x-b}{a}},\frac{y-\sqrt{y^2-a^2}}{a}=e^{-\frac{x-b}{a}}.$$

Adding these yields

$y={\displaystyle \frac{a}{2}}\left(e^{\frac{x-b}{a}}+e^{-\frac{x-b}{a}}\right)=a\cosh {\displaystyle \frac{x-b}{a}}.$ (3)

From this we see that the extremals $c$ of the problem are catenaries.  It means that the least surface of revolution in the question is a catenoid.

Title	least surface of revolution
Canonical name	LeastSurfaceOfRevolution
Date of creation	2013-03-22 19:12:11
Last modified on	2013-03-22 19:12:11
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Example
Classification	msc 49K05
Classification	msc 53A05
Classification	msc 26B15
Related topic	MinimalSurface
Related topic	EquationOfCatenaryViaCalculusOfVariations
Related topic	Catenary
Related topic	MinimalSurface2
Related topic	CalculusOfVariations
Related topic	SurfaceOfRevolution2