\begin{definition}[Definition:Mutually Consistent Boundary Conditions/wrt Functional]
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 {{WRT}} the Euler equations, then these boundary conditions are called '''mutually consistent''' {{WRT}} the functional $J$.
\end{definition}