Necessary and Sufficient Condition for First Order System to be Field for Second Order System

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

Let $ \boldsymbol \psi $ be continuously differentiable.

Then $ \forall x \in \left [ { a \,. \,. \, b } \right ] $ the first-order system


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

is a field for the second-order system


 * $ \mathbf y'' = \mathbf f \left ( { x, \mathbf y, \mathbf y' } \right ) $

iff $ \boldsymbol \psi $ satisfies


 * $ \displaystyle \frac{ \partial \boldsymbol \psi }{ \partial x } + \sum_{ i = 1 }^N \frac{ \boldsymbol \psi }{ \partial y_i } \psi_i = \mathbf f \left ( { x, \mathbf y, \boldsymbol \psi } \right ) $

In other words, every solution to Hamilton-Jacobi system is a field for the original system.

Necessary condition
Differentiate the first-order system $ x $:

Denote:


 * $ \displaystyle \frac{ \mathrm d y_i }{ \mathrm d x } = \psi_i \left ( { x, \mathbf y } \right ) \left ( { \star } \right ) $

Suppose:


 * $ \displaystyle \frac{ \partial \boldsymbol \psi }{ \partial x } + \sum_{ i = 1 }^N \frac{ \partial \boldsymbol \psi }{ \partial y_i } \psi_i = \mathbf f \left ( { x, \mathbf y, \mathbf y' } \right ) $

Then


 * $ \mathbf y'' = \mathbf f \left ( { x, \mathbf y, \mathbf y' } \right ) \left ( { \ast } \right ) $

By assumption, the first-order system $ \left ( { \star } \right ) $ is valid in the interval $ \left [ { a \,. \,. \, b } \right ] $.

By definition, it is a field for the second-order system $ \left ( { \ast } \right ) $.