# Partial Derivatives of Solution of Hamilton-Jacobi Equation are First Integrals of Euler's Equations

## Theorem

Let $\mathbf y=\sequence{y_i}_{1\le i\le n}$, $\boldsymbol\alpha=\sequence{\alpha_i}_{1\le i\le m}$ be vectors, where $m\le n$.

Let $S=\map S {x,\mathbf y,\boldsymbol\alpha}$ be a solution of Hamilton-Jacobi quation, where $ \boldsymbol \alpha$ are parameters.

Then each derivative

- $\displaystyle\frac{\partial S}{\partial\alpha_i}$

is a first integral of canonical Euler's equations.

## Proof

Consider the total derivative of $\displaystyle\frac{\partial S}{\partial\alpha_i}$ wrt $x$:

\(\displaystyle \frac \d {\d x}\frac{\partial S}{\partial\alpha_i}\) | \(=\) | \(\displaystyle \frac{\partial^2 S}{\partial x\partial\alpha_i}+\sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\frac{\d y_j}{\d x}+\sum_{j=1}^m+\frac{\partial^2 S}{\partial\alpha_j\partial\alpha_i}\frac{\d\alpha_j}{\d x}\) | Total derivative of $S$ wrt $x$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac{\partial^2 S}{\partial x\partial\alpha_i}+\sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\frac{\d y_j}{\d x}\) | $ \alpha_i$ is parameter, independent of $x$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\frac{\partial H}{\partial\alpha_i}+\sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\frac{\d y_j}{\d x}\) | $S$ satisfies Hamilton-Jacobi equation | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\frac{\partial x}{\partial\alpha_i}\frac{\partial H}{\partial x}-\sum_{j=1}^n\frac{\partial y_j}{\partial\alpha_i}\frac{\partial H}{\partial y_j}-\sum_{j=1}^n\frac{\partial p_j}{\partial\alpha_i}\frac{\partial H}{\partial p_j}+\sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\frac{\d y_j}{\d x}\) | Partial derivative of multivariate composite function $\map H {x,\mathbf y,\mathbf p}$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\frac{\partial H}{\partial p_j}+\sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\frac{\partial y_j}{\partial x}\) | $x$, $y_j$ independent of $\alpha_i$; $S$ satisfies Hamilton-Jacobi equation, thus $\displaystyle p_j=\frac{\partial S}{\partial y_j}$ | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{j=1}^n\frac{\partial^2 S}{\partial y_j\partial\alpha_i}\paren{\frac{\d y_j}{\d x}-\frac{\partial H}{\partial p_j} }\) |

If Euler's equations are satisfied, RHS vanishes.

Hence

- $\displaystyle\frac \d {\d x} \frac{\partial S}{\partial\alpha_i}=0$

or

- $\displaystyle\frac{\partial S}{\partial\alpha_i}=C_i$

where $C_i$ is a constant.

$\blacksquare$

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.23$: The Hamilton-Jacobi Equation. Jacobi's Theorem