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# isoperimetric problem

The simplest of the isoperimetric problems is the following:

*One must set an arc with a given length $l$ from a given point $P$ of the plane to another given point $Q$ such that the arc together with the line segment $PQ$ encloses the greatest area possible.*

This task is solved in the entry example of calculus of variations.

More generally, *isoperimetric problem* may mean determining such an arc $c$ between the given points $P$ and $Q$ that it gives for the integral

$\displaystyle\int_{P}^{Q}\!f(x,\,y,\,y^{{\prime}})\,ds$ | (1) |

an extremum and that gives for another integral

$\displaystyle\int_{P}^{Q}\!g(x,\,y,\,y^{{\prime}})\,ds$ | (2) |

a given value $l$, as both integrals are taken along $c$. Here, $f$ and $g$ are given functions.

The constraint (2) can be omitted by using the function $f\!-\!\lambda g$ instead of $f$ in (1) similarly as in the mentionned example.

## Mathematics Subject Classification

47A60*no label found*49K22

*no label found*

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