Definition:Self-Adjoint Boundary Conditions

Definition
Consider the functional $ J \left [ { \mathbf y } \right ] $, such that


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

Let the momenta of $ J $ be:


 * $ \displaystyle \mathbf p = \nabla_{ \mathbf y' } F \left ( { x, \mathbf y, \mathbf y' } \right ) $

Let the following boundary conditions hold:


 * $ \mathbf y' \left ( { a } \right ) = \boldsymbol \psi \left ( { \mathbf y } \right ) \vert_{ x = a } $

If


 * $ \exists g \left ( { x, \mathbf y } \right ) : \mathbf p \left ( { x, \mathbf y, \boldsymbol \psi \left ( { \mathbf y } \right ) } \right ) \vert_{ x = a } = \nabla_{ \mathbf y' } g \left ( { x, \mathbf y } \right ) \vert_{ x = a }$

then the boundary conditions are called self-adjoint.