# 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.

## 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

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 6.31$: Consistent Boundary Conditions. General Definition of a Field