Conditions for Extremal Embedding in Field of Functional

Theorem
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 $

Let $ \gamma $ be an extremal of $ J $, defined by $ \mathbf y = \mathbf y \left ( { x } \right ) $ for $ x \in \left [ { a \,. \,. \, b } \right ] $.

Suppose:


 * $ \det \left ( { F_{ \mathbf y' \mathbf y' } } \right ) \ne 0 \quad \forall x \left [ { a \,. \,. \, b } \right ] $

Suppose no points conjugate to $ \left ( { a, \mathbf y \left ( { a } \right ) } \right ) $ lie on $ \gamma $.

Then $ \gamma $ can be imbedded in a field.

Proof
By assumption there exists $ \epsilon > 0 $ such that:


 * $ \gamma $ can be extended onto the whole interval $ \left [ { a - \epsilon \,. \,. \, b } \right ] $


 * the interval $ \left [ { a - \epsilon \,. \,. \, b } \right ] $ contains no points conjugate to $ a $.

Consider a family of extremals leaving the point $ \left ( { a - \epsilon, \mathbf y \left ( { a - \epsilon } \right ) } \right ) $.

There are no points conjugate to $ a - \epsilon $ in $ \left [ { a - \epsilon \,. \,. \, b } \right ] $.

Hence, for $ x \left [ { a \,. \,. \, b } \right ] $ no two extremals in this family which are sufficiently close to the original extremal $ \gamma $ can intersect.

Thus, in some region $ R $ containing $ \gamma $ extremals sufficiently close to $ \gamma $ define a central field in which $ \gamma $ is imbedded.

By Central Field is Field of Functional, $ \gamma $ can be imbedded in the field of functional.