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:
$$ \frac{\partial p_i}{\partial t} = -\frac{\partial H}{\partial q_i}, \quad \frac{\partial q_i}{\partial t} = \frac{\partial H}{\partial p_i}. $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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