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# classical isoperimetric problem

The points $a$ and $b$ on the $x$-axis have to be connected by an arc with a given length $l$ such that the area between the $x$-axis and the arc is as great as possible.

Denote the equation of the searched arc by $y=y(x)$. The task, which belongs to the isoperimetric problems, can be formulated as

$\displaystyle\mbox{to maximise}\quad\int_{a}^{b}\!y\,dx$ | (1) |

under the constraint condition

$\displaystyle\int_{a}^{b}\!\sqrt{1\!+\!y^{{\prime\,2}}}\,dx\;=\;l.$ | (2) |

We have the integrands

$f(x,\,y,\,y^{{\prime}})\;\equiv\;y,\quad g(x,\,y,\,y^{{\prime}})\;\equiv\;% \sqrt{1\!+\!y^{{\prime\,2}}}.$ |

The conditional variation problem for the functional in (1) may be considered as a free variation problem (without conditions) for the functional $\int_{a}^{b}(f\!-\!\lambda g)\,dx$ where $\lambda$ is a Lagrange multiplier. For this end we need the Euler–Lagrange differential equation

$\displaystyle\frac{\partial}{\partial y}(f\!-\!\lambda g)-\frac{d}{dx}\frac{% \partial}{\partial y^{{\prime}}}(f\!-\!\lambda g)\;=\;0.$ | (3) |

Since the expression $f\!-\!\lambda g$ does not depend explicitly on $x$, the differential equation (3) has, by the Beltrami identity, a first integral of the form

$(f\!-\!\lambda g)-y^{{\prime}}\!\cdot\!(f^{{\prime}}_{{y^{{\prime}}}}\!-\!% \lambda g^{{\prime}}_{{y^{{\prime}}}})\;=\;C_{2},$ |

which reads simply

$y-\frac{\lambda}{\sqrt{1\!+\!y^{{\prime\,2}}}}\;=\;C_{2}.$ |

This differential equation may be written

$y^{{\prime}}\;\equiv\;\frac{dy}{dx}\;=\;\frac{\sqrt{\lambda^{2}\!-\!(y\!-\!C_{% 2})^{2}}}{y\!-\!C_{2}},$ |

where one can separate the variables and integrate, obtaining the equation

$(x\!-\!C_{1})^{2}+(y\!-\!C_{2})^{2}\;=\;\lambda^{2}$ |

of a circle. Here, the parametres $C_{1},\,C_{2},\,\lambda$ may be determined from the conditions

$y(a)\;=\;y(b)\;=\;0,\quad\mbox{arc length}\;=\;l.$ |

Thus the extremal of this variational problem is a circular arc 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.

## Mathematics Subject Classification

47A60*no label found*49K22

*no label found*49K05

*no label found*

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