# 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

 $\displaystyle A\;=\;2\pi\int_{P_{1}}^{P_{2}}\!y\,ds,$ (1)

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

 $\displaystyle\int_{P_{1}}^{P_{2}}\!y\,ds\;=\;\int_{x_{1}}^{x_{2}}\sqrt{1\!+\!y% ^{\prime 2}}\,|dx|.$ (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

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

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

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

i.e.

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

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