# What makes an equation an 'equation of motion'?

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?

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I would not be surprised if physicists are often imprecise about the terminology. –  David Z Dec 3 '12 at 18:52

In general, a dynamical equation of motion or evolution equation is a (hyperbolic) second order in time differential equation. They determine the evolution of the system.

$\partial_{\mu}F^{i\mu}$ is a dynamical equation.

However, a constraint is a condition that must be verified at every time and, in particular, the initial conditions have to verify the constraints. Since equations of motion are of order two in time, constraints have to be at most order one.

The Gauss law $\partial_{\mu}F^{0\mu}$ is a constraint because it only involves a first derivative in time in configuration space, i.e., when $\bf E$ it is expressed in function of $A_0$ and $\bf A$. Furthermore, the gauss law is the generator of gauge transformations. In the quantum theory, only states which are annihilated by the gauss law are physical states.

Both dynamical equations and constraints may be called equations of motion or Euler-Lagrange equations of a given action functional. Or, one may keep the term equation of motion for dynamical equations. It is a matter of semantic. The important distinction is between constraints and evolution equations.

Conservation laws follow mainly from symmetries and from Noether theorem. Often but not always, equations of motion follow from conservation laws. Whether one considerers one more fundamental is a matter of personal taste.

Dirac equation relates several components of a Dirac spinor. Each component verifies the Klein-Gordon equation which is an evolution equation of order two.

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An equation of motion is a (system of) equation for the basic observables of a system involving a time derivative, for which some initial-value problem is well-posed.

Thus a continuity equation is normally not an equation of motion, though it can be part of one, if currents are basic fields.

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In field theory, a conservation law just states that some quantity is conserved: if $\partial_\mu \, \star = 0$ where $\star$ is a vector or a tensor, you can associate a conserved charge etc. - you know the spiel I guess.

Constraints are something you impose by hand (or by experiment).

Finally, equations of motion are dynamical equations that follow from the Euler-Lagrange equation. Both the Dirac equation and $\partial^\mu F_{\mu \nu} = 0$ satisfy that criterion. [However, if you choose a gauge, it becomes much more clear that you're dealing with a PDE for the field $A_\mu$, such as $\square A_\mu = 0.$] Also note that both of them involve $\partial_t$ or $\partial_t^2$. Note that $\nabla \cdot \mathbf{E}$ only involves spatial derivatives, so it gives no dynamics.

In your example, $\partial^\mu j_\mu = 0$ is a classical conservation law that doesn't describe how some microscopic quantity evolves in time - you don't derive it from Euler-Lagrange.

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OP wrote(v2):

What makes an equation an 'equation of motion'?

As David Zaslavsky mentions in a comment, in full generality, there isn't an exact definition. Loosely speaking, equations of motion are evolution equations, with which the dynamical variables future (and past) behavior can be determined.

However, if a theory has an action principle, then there is a precedence within the physics community, see e.g. Ref. 1. Then only the Euler-Lagrange equations are traditionally referred to as 'equations of motion', whether or not they are dynamical equations (i.e. contain time derivatives) or constraints (i.e. do not contain time derivatives).

References:

1. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.
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Is there any new in your answer with respect to mine? I do not mind votes, I just want to know whether I do not express my ideas clearly. –  drake Dec 4 '12 at 17:54
Could you answer my question? –  drake Dec 5 '12 at 19:51
Dear @drake: Your answer(v2) mentions e.g. explicitly order two in various places, which mine doesn't. OP's question is a soft terminology question, where people in different fields may use the term slightly differently. In such situations, it may be good to cite a reference or two. –  Qmechanic Dec 5 '12 at 20:19
Dear Qmechanic, thank you for answering. I wrote "second order" and "first order" because, as I wrote in my answer, I am in configuration space. You wrote "contain time derivative" and "do not contain time derivative" because you are thinking of phase space. OK, about the references. –  drake Dec 5 '12 at 20:31