# 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

$A=\mathrm{\hspace{0.33em}2}\pi {\displaystyle {\int}_{{P}_{1}}^{{P}_{2}}}y\mathit{d}s,$ | (1) |

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

${\int}_{{P}_{1}}^{{P}_{2}}}y\mathit{d}s={\displaystyle {\int}_{{x}_{1}}^{{x}_{2}}}\sqrt{1+{y}^{\prime 2}}|dx|.$ | (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

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

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

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