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 $\forall x\in\closedint a b$ the first-order system


 * $\mathbf y'=\map{\boldsymbol\psi} {x,\mathbf y}$

is a field for the second-order system


 * $\mathbf y''=\map{\mathbf f} {x,\mathbf y,\mathbf y'}$

iff $\boldsymbol\psi$ satisfies


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

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

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

This can be rewritten as the following system of equations:


 * $\mathbf y''=\map{\mathbf f} {x,\mathbf y,\mathbf y'}$


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

By assumption, the first-order system is valid in $\closedint a b$.

For the second-order system to be valid in the same interval, the corresponding Hamilton-Jacobi equation has to hold as well.