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I come across the terms "equations of motion" and "field equations" all the time, but what is the difference? For example, general relativity is described in terms of the Einstein field equation $G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu}$ and particle dynamics are governed by the geodesic equation (of motion) $\frac{d^2\gamma^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{d\gamma^\mu }{dt}\frac{d\gamma^\nu }{dt} = 0.$ There is a similar case for electromagnetism: one might say the field equations are Maxwell's equations and the equation of motion is the Lorentz force law.

But what is the precise distinction between these two terms? For example, if I am interested in the simple scalar field theory $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2$$ and I use the Euler-Lagrange equations to derive $$(\partial_\mu\partial^\mu+m^2)\phi=0$$ would one refer to this as a field equation, an equation of motion, or both?

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There is no "precise distinction" between these terms. The "field equations" are just important equations of a field theory, which may or may not be the equations of motion for that theory.

And even which equations are "equations of motion" is not unique! In the Lagrangian formulation of electromagnetism coupled to a charged current, you get two of Maxwell's four equations as its equations of motion, but the other two are just an abstract constraint on the field strength tensor. And if you take the Lagrangian theory of a charged particle coupled to a background EM field, then you get the Lorentz force law as the equation of motion, and no Maxwell equations at all.

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Field equations tell you how fields change in spacetime, whereas equations of motion tell you how arbitrary physical objects move in spacetime. In other words, field equations are equations of motion for a field.

Normally the term EOM is used in classical mechanics to denote the motion of a single body or system of bodies (though it certainly is used in quantum theories), whereas field equation is used often in both classical and quantum theories.

What you have written there is the Klein-Gordon equation, and yes it is a field equation for a "free massive scalar field" in QFT. It tells you the EOM for the field. If you couple that Lagrangian to other fields or to particles you can get other EOMs and field equations.

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  • Equations of motion (EOM) are typically the equations that determine the time-evolution of the system.

    E.g. in Newtonian mechanics, Newton's 2nd law is the EOM. (One should avoid referring to the kinematic $suvat$-equations as EOM to avoid confusion.)

  • For Lagrangian systems, the Euler-Lagrange (EL) equations are referred to as EOM, even in case of EL equations for non-dynamical variables.

  • Let us now return to OP's title question. In field theory (as opposed to point mechanics), the EOMs are also called field equations.

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The practical (although probably not the most rigorous) definition comes from looking at the derivatives in the equation.

Equations of motion describe the evolution of the position of particles, so the independent variable is time (or a parameter like proper time) and the dependent variables are spatial position (or spacetime event locations). You'll see a differential equation that has only 1 type of derivative, and no partial derivatives.

Field equations describe the evolution of fields - the independent variables describe spatial and temporal domains and the dependent variables are fields that take values at each point in those domains. The time derivative is usually only one of the several partial derivatives that appear in the equation.

Your scalar field equation is a field equation because $phi$ is something defined over a multidimensional domain, as indicated by the partial derivatives.

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