Definition:Mutually Consistent Boundary Conditions/wrt Functional

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


 * $ \displaystyle J = \int_a^b F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x$

where its Euler's equations are:


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

Consider the following boundary conditions:


 * $ \displaystyle \mathbf y\vert_{ x = x_1 } = \boldsymbol \psi^{ \left ( { 1 } \right ) } \left ( { \mathbf y } \right ) \vert_{ x = x_1 } $


 * $ \displaystyle \mathbf y\vert_{ x = x_2 } = \boldsymbol \psi^{ \left ( { 2 } \right ) } \left ( { \mathbf y } \right ) \vert_{ x = x_2 } $

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