Definition:Hamilton-Jacobi Equation

Definition
Let $S \left({ x_0, x_1, \mathbf y } \right)= S \left({ x, \mathbf y } \right)$ 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} + H \left({ x, \mathbf y, \nabla_{ \mathbf y}S } \right)=0$

is known as the Hamilton-Jacobi Equation.