Given two points in the Plane, find the curve which minimizes the distance between them. The Line Element
is given by

(1) |

(2) |

(3) |

(4) | |||

(5) |

so the Euler-Lagrange Differential Equation becomes

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

(12) | |||

(13) |

Writing (12) and (13) as a Matrix Equation gives

(14) |

(15) |

so

(16) | |||

(17) |

(18) |

as expected.

The shortest distance between two points on a Sphere is the so-called Great Circle distance.

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 930-931, 1985.

© 1996-9

1999-05-25