Definition:Mutually Consistent Boundary Conditions/wrt Functional

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 the Euler equations, then these boundary conditions are called mutually consistent  the functional $J$.