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 \in \left [ { a \,. \,. \, b } \right ] $

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

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

Proof
Let $c \in \R$ be conjugate to $a$, such that $c < a$.

By assumption:


 * $c \notin \left [ { a \,. \,. \, b } \right ] $.

Hence, there exists a set $ \left [ { c \,. \,. \, b } \right ] $:


 * $ \left [ { c \,. \,. \, b } \right ] =  \left [ { c \,. \,. \, a } \right ) \cup  \left [ { a \,. \,. \, b } \right ] $

where $\left\vert{c - a}\right \vert > 0$.

By there exists a real point between two real points:


 * $\exists \epsilon: \left\vert{c - a}\right\vert > \epsilon > 0$

Therefore, there exists $ \epsilon > 0 $ such that:


 * $\gamma$ can be extended onto the whole interval $ \left [ { a - \epsilon \,. \,. \, b } \right ] $, where extension means definition of some mapping in $ \left [ { a-\epsilon \,. \,. \, a } \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 \in \left [ { a \,. \,. \, b } \right ] $ no two extremals in this family which are sufficiently close to the original extremal $\gamma$ can intersect.

Since all the functions are extremals, they satisfy same differential equations.

Lack of intersection implies different boundary conditions.

Denote these conditions collectively as $ \boldsymbol \psi \left ( { x, \mathbf y } \right ) = \mathbf y' \left ( { x } \right ) $.

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

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