Derivation of Hamilton-Jacobi Equation

Theorem
Let $\map S {x_0,x_1,\mathbf y}=\map S {x,\mathbf y}$ be the geodetic distance, where $x_0$ is fixed and $x_1=x$.

Let $H$ be Hamiltonian.

Then the following equation holds:


 * $\dfrac {\partial S} {\partial x}+\map H {x,\mathbf y,\nabla_{\mathbf y}S}=0$

and is known as the Hamilton-Jacobi Equation.

Proof
Consider the increment $\Delta S$:


 * $\Delta S=\map S {x+\Delta x,\mathbf y+\Delta\mathbf y}-\map S {x,\mathbf y}$

Note that the change of function $\mathbf y$ denoted by $\Delta \mathbf y$ is dependent on the manner $\Delta x$ is chosen through the definition of geodetic distance.

For sufficiently smooth $S$, $\size{\Delta\mathbf y}\to 0$ as $\size{\Delta\mathbf x}\to 0$.

By definition of differential, $\Delta S$ can be written as


 * $\map {\Delta S} {x,\mathbf y;\Delta x,\Delta\mathbf y}=\rd \map S {x,\mathbf y;\Delta x,\Delta\mathbf y}+\epsilon\Delta x+\boldsymbol\epsilon\cdot\Delta\boldsymbol y$

where $\epsilon\to 0$ as $\Delta x\to 0$, and $\size{\mathbf y}\to 0$ as $\size{\Delta\mathbf x}\to 0$.

By definition of the geodetic distance,


 * $\Delta S=J\sqbrk{\gamma^*}-J\sqbrk{\gamma}$

where $\gamma$ and $\gamma^*$ are extremal curves, connecting the fixed initial point with points $\paren{x,\mathbf y}$ and $\paren{x+\Delta x,\mathbf y+\mathbf h}$ respectively.

By definition of increment of functional:


 * $J\sqbrk{\gamma^*}-J\sqbrk{\gamma}=\Delta J\sqbrk{\gamma;\Delta\gamma}$

where $\Delta\gamma=\gamma^*-\gamma$.

A differentiable $J$ can be expressed as:


 * $\Delta J\sqbrk{\gamma;\Delta\gamma}=\delta J\sqbrk{\gamma;\Delta\gamma}+\epsilon_\gamma\cdot\size{\Delta\gamma}$

where $\epsilon_\gamma\to 0$ as $\size{\Delta\gamma}\to 0$, and $\size{\Delta\gamma}\to 0$ as $\size{\Delta\mathbf x}\to 0$ for sufficiently smooth $S$.

To summarise:


 * $\Delta \map S {x,\mathbf y;\Delta x,\Delta\mathbf y}=\Delta J\sqbrk{\gamma;\Delta\gamma}$

Both sides contain terms linear in $\size{\Delta x}$, $\size{\Delta\mathbf y}$, $\size{\Delta\gamma}$ as well terms of higher order.

Higher order terms on both sides are the same.

Hence, the principal parts match:


 * $\rd S=\delta J$

The variation of extremal $J$ is expressible as


 * $\displaystyle\delta J=\sum_{i=1}^n p_i\mathrm \Delta y_i-H\mathrm\Delta x$

while, the differential of $S$ is


 * $\displaystyle \rd S=\frac{\partial S}{\partial x}\Delta x+\sum_{i=1}^n\frac{\partial S}{\partial y_i}\Delta y_i$

Equivalently:


 * $\displaystyle \paren{\frac{\partial S}{\partial x}+H }\Delta x+\sum_{i=1}^n\paren{\frac{\partial S}{\partial y_i}-p_i}\Delta y_i=0$

$\Delta x$ and $\Delta y_i$ are independent variables.

The equation holds only if all the coefficients in front of $\Delta x$ and $\Delta y_i$ vanish simultaneously:


 * $\dfrac {\partial S} {\partial x}=-H,\quad\dfrac {\partial S} {\partial y_i}=p_i$

Since $H=\map H {x,\mathbf y,\mathbf p}$, using the second relation to replace $\mathbf p$ together with the first one proves the formula.