Hamilton-Jacobi equations

In physics and mathematics, the Hamilton-Jacobi equations are equations of classical physics that describe the motion of a physical object defined by an energy functional. The solutions of the Hamilton-Jacobi equations are the integral curves of the Hamiltonian vector field on a symplectic manifold. They are named after William Rowan Hamilton and Carl Gustav Jacob Jacobi.

Definition

In canonical coordinates, the equations are:

$\dot{q}^i = \frac {\partial H}{\partial p_i}$

and

$\dot{p}_i = - \frac {\partial H}{\partial q^i}$ .

The solutions to these equations can be understood to be the integral curves of Hamiltonian vector fields on a symplectic manifold.