Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint.
For example, in the Hamiltonian formulation of Maxwell's theory, Gauss' law $\nabla\cdot\mathbf{E}=0$ is a constraint, whereas $\partial_\mu F^{\mu\nu}=0$ is an equation of motion. But why then isn't $\partial_\mu j^\mu=0$, the charge-conservation/continuity equation, called an equation of motion. Instead it is just a 'conservation law'.
Maybe first-order differentials aren't allowed to be equations of motion? Then what of the Dirac equation $(i\gamma^\mu\partial_\mu-m)\psi=0$? This is a first-order differential, isn't it? Or perhaps when there is an $i$, all bets are off...
So, what counts as an equation of motion, and what doesn't? How can I tell if I am looking at a constraint? or some conservation law?