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 complete solution of Hamilton-Jacobi equation.

Let:


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

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.

Proof 1
Consider the total derivative of $\displaystyle\frac{\partial S}{\partial\alpha_i}$ $x$:

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

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

By comparison of this and previous expressions:


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

Proof 2
Consider canonical Euler's equations:


 * $\displaystyle\frac{\d y_i}{\d x}=\frac{\partial H}{\partial p_i},\quad\frac{\d p_i}{\d x}=-\frac{\partial H}{\partial y_i}$

Apply a canonical transformation $\paren{x,\mathbf y,\mathbf p,H}\to\paren{x,\boldsymbol \alpha,\boldsymbol\beta,H^*}$, where $\Phi=S$.

By Conditions for Transformation to be Canonical:


 * $\displaystyle p_i=\frac{\partial S}{\partial y_i},\quad\beta_i=\frac{\partial S}{\partial\alpha_i},\quad H^*=H+\frac{\partial S}{\partial x}$

Since $S$ satisfies Hamilton-Jacobi equation, $ H^*=0$.

In these new coordinates canonical Euler's equations are:


 * $\displaystyle\frac{\d\alpha_i}{\d x}=\frac{\partial H^*}{\partial\beta_i}$


 * $\displaystyle \frac{\d\beta_i}{\d x}=-\frac{\partial H^*}{\partial\alpha_i}$

By $H^*=0$:


 * $\displaystyle\frac{\d\alpha_i}{\d x}=0,\quad\displaystyle\frac{\d\beta_i}{\d x}=0$

which imply that $ \alpha_i$ and $\beta_i$ are constant along each extremal.

$\beta_i$ constancy provides with $n$ first integrals:


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

Because $S=\map S {x,\mathbf y,\boldsymbol\alpha}$, the aforementioned set of first integrals is also a system of equations for functions $y_i$.

Thus, functions $y_i$ can be found.

Functions $p_i$ are found by the results of Conditions for Transformation to be Canonical:


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

Then


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


 * $\map {p_i} {x, \boldsymbol \alpha, \boldsymbol \beta}$

are solutions to canonical Euler's equations.