# Definition:Geodetic Distance

## Definition

Let $y_i$, $F$ be real functions.

Let $\mathbf y=\sequence{y_i}_{1\le i\le n}$ be a vector.

Let

$\displaystyle J\sqbrk{\mathbf y}=\int_{x_0}^{x_1}\map F {x,\mathbf y,\mathbf y'}\rd x$

be a functional, which has only one extremal passing any two points

$A=\map A {x_0,\mathbf y^0},\quad B=\map B {x_1,\mathbf y^1}$

Suppose a curve $\gamma$ is an extremal of $J$.

Then

$\displaystyle \map S {x_0,x_1,\mathbf y}=\int_{x_0}^{x_1}\map F {x,\mathbf y,\mathbf y'}\big\rvert_{\gamma}\rd x$

is called a geodetic distance between $A$ and $B$.