# Definition:Geodetic Distance

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## 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 $B$.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.23$: The Hamilton-Jacobi Equation. Jacobi's Theorem