# Definition:Geodetic Distance

## Definition

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

Let $\mathbf y = \sequence {y_i}_ {1 \mathop \le i \mathop \le n}$ 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 with only one extremal passing any two points:

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

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 $B$.