Necessary and Sufficient Condition for First Order System to be Mutually Consistent

Theorem
Let $ \mathbf y $, $ \boldsymbol \psi $ be N-dimensional vectors.

Let $ g $ be a twice differentiable mapping.

Let


 * $ \boldsymbol \psi \left ( { x, \mathbf y } \right ) = \mathbf y' \left ( { x, \mathbf y } \right) \quad \left ( { 1 } \right ) $


 * $ \mathbf p \left [ { x, \mathbf y, \boldsymbol \psi \left ( { x, \mathbf y } \right ) } \right ] = g_{ \mathbf y } \left ( { x, \mathbf y } \right ) \quad \left ( { 2 } \right ) $

where $ \mathbf p $ is a momentum.

Then the boundary conditions defined by $ \left ( { 1 } \right ) $ are mutually consistent iff the mapping $ g \left ( { x, \mathbf y } \right ) $ satisfies the Hamilton-Jacobi equation:


 * $ \displaystyle \frac{ \partial g }{ \partial x } + H \left ( { x, \mathbf y, \frac{ \partial g }{ \partial \mathbf y } } \right ) = 0 \quad \left ( { 3 } \right ) $

Necessary condition
Take the partial derivative of $ \left ( { 3 } \right ) $ $ x $:


 * $ \displaystyle \frac{ \partial^2 g \left ( { x, \mathbf y } \right ) }{ \partial \mathbf y \partial x } = - \frac{ \partial }{ \partial \mathbf y} H \left( { x, \mathbf y, \frac{ \partial g }{ \partial \mathbf y } } \right )

\quad \left ( { 4 } \right ) $

By Schwarz-Clairaut theorem, the order of partial derivatives of $ g $ can be exchanged:


 * $ \displaystyle \frac{ \partial^2 g \left ( { x, \mathbf y } \right ) }{ \partial \mathbf y \partial x } = \frac{ \partial \mathbf p \left [ { x, \mathbf y, \boldsymbol \psi \left ( { x, \mathbf y }\right ) } \right ] }{ \partial x } \quad \left ( { 5 } \right ) $

Since $ g \left ( { x, \mathbf y } \right ) $ depends on $ \mathbf y $ only in a direct way, $ \displaystyle \frac{ \partial g }{ \partial \mathbf y } = g_{ \mathbf y } $

Hence, from $ \left ( { 2 } \right ) $, $ \left ( { 4 } \right ) $ and $ \left ( { 5 } \right ) $ it follows that:


 * $ \displaystyle \frac{ \partial \mathbf p \left [ { x, \mathbf y, \boldsymbol \psi \left ( { x, \mathbf y }\right ) } \right ] }{ \partial x } = - \frac{ \partial }{ \partial \mathbf y}  H \left( { x, \mathbf y, \mathbf p \left [ { x, \mathbf y, \boldsymbol \psi \left ( { x, \mathbf y }\right ) } \right ] } \right ) $

These are the consistency equations.