# Definition:Extremal Embedding in Field of Functional

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## Definition

Let $J$ be a functional such that:

- $\displaystyle J\sqbrk{\mathbf y}=\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

Let $\gamma$ be an extremal of $J$.

Let $R$ be a simply connected open region which contains $\gamma$ as a subset.

Let a field of functional $J$ be defined at every point of $R$.

Let one of the trajectories of the field be $\gamma$.

Then $\gamma$ can be **embedded** in a field of functional $J$.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.32$: The Field of a Functional