Jacobi's Theorem/Proof 1

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

Next, consider the total derivative of $p_i$ $x$:

On the other hand, the partial derivative of Hamilton-Jacobi equation yields:

By comparison of this and previous expressions:


 * $\dfrac {\d p_i} {\d x} = -\dfrac {\partial H} {\partial y_i}$