Definition:Central Field
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Definition
Let $R$ be a simply connected region in $\paren {n + 1}$-dimensional space.
Let $\tuple {x, \mathbf y}$ be a point in $R$.
Let $c = \tuple {\sequence {c_i}_{0 \mathop \le i \mathop \le n} }$ be a point lying outside of $R$.
Let $J$ be a functional such that:
- $\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
whose extremals $\mathbf y$ are curves in $\paren{n+1}$-dimensional space.
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Suppose that one and only one extremal of $J$ leaves $c$ and passes through $\tuple {x, \mathbf y}$, thereby for every point in $R$ defining a relation:
- $(1): \quad \map {\mathbf y'} x = \map {\boldsymbol \psi} {x, \mathbf y}$.
Then the field of directions $(1)$ is called a central field.
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.32$: The Field of a Functional