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

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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.

$\blacksquare$

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