Hamiltonian equations

Given a Hamiltonian vector field $X_{F}$ , the equation governing its flow, i.e., the components of it in specific coordinates, are called Hamiltonian equations for $F$ . See @olver86 page 392.

If $F = H$ is the energy of a system (the Hamiltonian) then they are also called the equations of motion of the system.

Example: with the standard symplectic form or Poisson bracket we have:

\frac{\partial p_{i}}{\partial t} = - \frac{\partial H}{\partial q_{i}}, \frac{\partial q_{i}}{\partial t} = \frac{\partial H}{\partial p_{i}} .