Jacobi's Theorem

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

Let $ S= S \left({ x, \mathbf y, \mathbf \alpha } \right)$ be a complete solution of Hamilton-Jacobi equation.

Let the determinant


 * $ \big \lvert \frac{ \partial^2 S}{ \partial \alpha_i \partial y_k} \big \rvert \ne 0$

Then


 * $ p_i= \frac{ \partial S}{ \partial y}$

and


 * $ y_i= y_i \left({ x, \boldsymbol \alpha, \boldsymbol \beta } \right)$

defined by


 * $ \frac{ \partial S}{ \partial \alpha_i}= \beta_i$

constitute a general solution of the canonical Euler's equations.