Definition:Self-Adjoint Boundary Conditions

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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.


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