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 \quad \left ( { 1 } \right ) $

where its Euler's equations are:


 * $ \displaystyle \nabla_{ \mathbf y' } F - \frac{ \mathrm d }{ \mathrm d x } \nabla_{ \mathbf y} F = 0 \quad \left ( { 2 } \right ) $

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

If they are consistent the system of equations $ \left ( { 2 } \right ) $,then these boundary conditions are called mutually consistent  the functional $ \left ( { 1 } \right ) $.