# 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**.

## Source of Name

This entry was named for William Rowan Hamilton and Carl Gustav Jacob Jacobi.

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.23$: The Hamilton-Jacobi Equation. Jacobi's Theorem