# 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**.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.31$: Consistent Boundary Conditions. General Definition of a Field