Definition:Geodetic Distance

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

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

Let


 * $ \displaystyle J[ \mathbf y] = \int_{x_0}^{x_1} F \left({ x, \mathbf y, \mathbf y' } \right) \mathrm d x$

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


 * $A=A \left({ x_0, \mathbf y^0 } \right), \quad B=B \left({ x_1, \mathbf y^1 } \right)$

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

Then


 * $ \displaystyle S \left({ x_0, x_1 } \right)= \int_{x_0}^{x_1}  F \left({ x, \mathbf y, \mathbf y' } \right) \big \rvert_{ \gamma} \mathrm d x$

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