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.

Then the equation


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

is known as the Hamilton-Jacobi Equation.