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 ) $