Definition:Self-Adjoint Boundary Conditions
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Definition
Consider the functional $J \sqbrk {\mathbf y}$, such that:
- $\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$
Let the momenta of $J$ be:
- $\mathbf p = \nabla_{\mathbf y'} \map F {x, \mathbf y, \mathbf y'}$
Let the following boundary conditions hold:
- $\map {\mathbf y'} a = \bigvalueat {\map {\boldsymbol \psi} {\mathbf y} } {x \mathop = a}$
If:
- $\exists \map g {x, \mathbf y}: \bigvalueat {\map {\mathbf p} {x, \mathbf y, \map {\boldsymbol \psi} {\mathbf y} } } {x \mathop = a} = \bigvalueat {\nabla_{\mathbf y'} \map g {x, \mathbf y} } {x \mathop = a}$
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