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

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

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

Then each partial derivative:


 * $\dfrac {\partial S} {\partial \alpha_i}$

is a first integral of canonical Euler's equations.

Proof
Consider the total derivative of $\dfrac {\partial S} {\partial \alpha_i}$ $x$:

If Euler's equations are satisfied, the vanishes.

Hence


 * $\dfrac \d {\d x} \dfrac {\partial S} {\partial \alpha_i} = 0$

or:


 * $\dfrac {\partial S} {\partial \alpha_i} = C_i$

where $C_i$ is a constant.