Definition:Mutually Consistent Boundary Conditions/wrt Functional

Definition
Let $J$ be a functional, such that:


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

where its Euler's equations are:


 * $\displaystyle\nabla_{\mathbf y'}F-\frac \d {\d x}\nabla_{\mathbf y}F=0$

Consider the following boundary conditions:


 * $\displaystyle\mathbf y\vert_{x=x_1}=\map{\boldsymbol\psi^{\paren 1} } {\mathbf y}\vert_{x=x_1}$


 * $\displaystyle\mathbf y\vert_{x=x_2}=\map{\boldsymbol\psi^{\paren 2} } {\mathbf y}\vert_{x=x_2}$

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