Jacobi's Theorem

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

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

Let:


 * $\begin {vmatrix} \dfrac {\partial^2 S} {\partial \alpha_i \partial y_k} \end{vmatrix} \ne 0$

where $\begin {vmatrix} \cdot \end{vmatrix}$ is a determinant.

Let:


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

Then:


 * $p_i = \map {\dfrac {\partial S} {\partial y_i} } {x, \mathbf y, \boldsymbol \alpha}$


 * $y_i = \map {y_i} {x, \boldsymbol \alpha, \boldsymbol \beta}$

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