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

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

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.

Proof

Necessary condition

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

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

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.

$\Box$


Sufficient condition


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