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:

<math>\dot{q}^i = \frac {\partial H}{\partial p_i}<math>

and

<math>\dot{p}_i = – \frac {\partial H}{\partial q^i}<math>.

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

See also

• In control theory, see Hamilton-Jacobi-Bellman equation.
• WKB approximation