# 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 $P\mathit{}Q$ encloses the greatest area possible.*

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

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

${\int}_{P}^{Q}}f(x,y,{y}^{\prime})\mathit{d}s$ | (1) |

an extremum and that gives for another integral

${\int}_{P}^{Q}}g(x,y,{y}^{\prime})\mathit{d}s$ | (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.

Title | isoperimetric problem |
---|---|

Canonical name | IsoperimetricProblem |

Date of creation | 2013-03-22 19:12:01 |

Last modified on | 2013-03-22 19:12:01 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 9 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 47A60 |

Classification | msc 49K22 |

Related topic | IsoperimetricInequality |

Related topic | LagrangeMultiplier |