Definition:Central Field

Definition
Let $ R $ be a simply connected region in $ \left ( { n + 1 } \right ) $-dimensional space.

Let $ \left ( { x, \mathbf y } \right ) $ be a point in $ R $.

Let $ c = \left ( { \langle c_i \rangle_{ 0 \le i \le n} } \right )$ be a point lying outside of $ R $.

Let $ J $ be a functional such that:


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

whose extremals $ \mathbf y $ are curves in $ \left ( { n + 1 } \right ) $-dimensional space.

Suppose that one and only one extremal of $ J $ leaves $ c $ and passes through $ \left ( { x, \mathbf y } \right ) $, thereby for every point in $ R $ defining a relation:


 * $ \mathbf y' \left ( { x } \right ) = \boldsymbol \psi \left ( { x, \mathbf y } \right ) \quad \left ( { \star } \right ) $.

Then the field of directions $ \left ( { \star } \right ) $ is called a central field.