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:
- $\ds J \sqbrk {\mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \rd x$
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 $J$.
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Then:
- $\ds \map S {x_0, x_1, \mathbf y} = \int_{x_0}^{x_1} \map F {x, \mathbf y, \mathbf y'} \big \rvert_{\gamma}\rd x$
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