\begin{definition}[Definition:Hamilton-Jacobi Equation]
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'''.
{{NamedforDef|William Rowan Hamilton|name2 = Carl Gustav Jacob Jacobi|cat = Hamilton|cat2 = Jacobi}}
\end{definition}