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

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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 of differential equations:

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

is a field for the second-order system

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

if and only if $\boldsymbol \psi$ satisfies :

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


That is, every solution to Hamilton-Jacobi system is a field for the original system.


Proof

Necessary condition

Differentiate the first-order system with respect to $x$:

\(\displaystyle \mathbf y''\) \(=\) \(\displaystyle \frac {\d \boldsymbol \psi} {\d x}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \frac {\d y_i} {\d x}\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\partial \boldsymbol \psi} {\partial x} + \sum_{i \mathop = 1}^N \frac {\partial \boldsymbol \psi} {\partial y_i} \psi_i\)

This can be rewritten as the following system of equations:

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

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.

$\Box$


Sufficient condition


Sources