Definition:Hamilton-Jacobi Equation

Definition
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.

The differential equation:


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

is known as the Hamilton-Jacobi Equation.