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 $x$:

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.