Definition:Self-Adjoint Boundary Conditions

Definition
Consider the functional $J\sqbrk{\mathbf y}$, such that


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

Let the momenta of $J$ be:


 * $\mathbf p=\nabla_{\mathbf y'}\map F {x,\mathbf y,\mathbf y'}$

Let the following boundary conditions hold:


 * $\map{\mathbf y'} a=\map{\boldsymbol\psi} {\mathbf y}\vert_{x=a}$

If:


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

then the boundary conditions are called self-adjoint.