Jacobi's Theorem/Proof 2

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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.

Proof

Consider canonical Euler's equations:

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

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

Apply a canonical transformation:

$\tuple {x, \mathbf y, \mathbf p, H} \to \tuple {x, \boldsymbol \alpha, \boldsymbol \beta, H^*}$

where $\Phi = S$.

By Conditions for Transformation to be Canonical:

$p_i = \dfrac {\partial S} {\partial y_i}$

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

$H^* = H + \dfrac {\partial S} {\partial x}$

Because $S$ is a solution to the Hamilton-Jacobi equation:

$H^* = 0$

In these new coordinates canonical Euler's equations are:

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

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

By $H^* = 0$:

$\dfrac {\d \alpha_i} {\d x} = 0$

$\dfrac {\d\beta_i} {\d x} = 0$

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

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

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

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.

$\blacksquare$

Source of Name

This entry was named for Carl Gustav Jacob Jacobi.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.23$: The Hamilton-Jacobi Equation. Jacobi's Theorem