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

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

Let $ S= S \left({ x, \mathbf y, \boldsymbol \alpha } \right)$ 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{ \mathrm d}{ \mathrm 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.