Definition:Mutually Consistent Boundary Conditions/wrt Functional

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Definition

Let $J$ be a (real) functional, such that:

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

where its Euler's equations are:

$\nabla_{\mathbf y'} F - \dfrac \d {\d x} \nabla_{\mathbf y} F = 0$

Consider the following boundary conditions:

$\bigvalueat {\mathbf y} {x \mathop = x_1} = \bigvalueat {\map {\boldsymbol \psi^{\paren 1} } {\mathbf y} } {x \mathop = x_1}$

$\bigvalueat {\mathbf y} {x \mathop = x_2} = \bigvalueat {\map {\boldsymbol \psi^{\paren 2} } {\mathbf y} } {x \mathop = x_2}$

If they are consistent with respect to the Euler equations, then these boundary conditions are called mutually consistent with respect to the functional $J$.

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