# What does the term 'equation of motion' refer to?

What does the term equation of motion refer to? If I am asked a question of the form 'What is the equation of motion of this object?', what should I write?

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Motion equation normally refers to something like $\vec x (t)$, but I don't know if that's what you are searching for - due to the vast amount of information you are providing about your problem... ;) – Benedikt Jun 2 '14 at 8:20
@Benedikt hehe i didnt provide enough information because I dont understand the term only. – user47593 Jun 2 '14 at 8:25
@user47593 Try this: physics.info/motion-equations or wikipedia en.wikipedia.org/wiki/Equations_of_motion – Anne O'Nyme Jun 2 '14 at 8:44
Do you really mean "equation of motion" as you wrote in the title, or "motion equation" as you wrote in the body? Whichever one you mean, you should edit the other to match. Emilio has answered the former case (where you mean "equation of motion" and thus you should fix the body) – David Z Jun 2 '14 at 9:03
@DavidZ Note that I edited the title for clarity, which read 'motion equation' originally. I see the two as equivalent as far as the latter is used, which is not very much; since the OP accepted my answer it seems to be fine. – Emilio Pisanty Jun 2 '14 at 14:12

The term 'equation of motion' is somewhat subjective as it depends on the context, but for any given context there is usually one single equation, or set of equations, which can be described as an equation of motion. These are typically differential equations in time, usually of second order, and for simple objects in Newtonian mechanics they do not involve other partial derivatives. In that context, equations of motion are usually expressions of Newton's 2nd law of motion.

Most importantly, the defining characteristic of an equation of motion is that its solutions, once appropriate initial conditions are fixed, must completely determine the evolution of the system after the initial time.

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I'd like to point out just one more case that hasn't been mentioned.

In Quantum Mechanics, the term "equation of motion" is (so far as I have seen) used to describe a differential equation that operators satisfy.

Let a system be described by a Hamiltonian $H$, and let there be a hermitian operator $A$ (an observable). Then, when working in the Heisenberg Picture (where the operators evolve in time and the kets are constant), the equation of motion of observable $A$ is:

$$\frac{dA^{(H)}}{dt}=\frac{1}{i\hbar}[A^{(H)},H^{(H)}],$$ where the $[,]$ denotes the conmutator and the superscript $(H)$ tells us that we're working in the Heisenberg Picture. One can also find an identical equation when working on the Interaction Picture.

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Well I can add one more thing. Having a Lagrangian $\mathcal L$ for system (I'm talking right now about classical mechanics) I can call Equations of Motion this:

$$\frac{\mathbf{d}}{\mathbf{d}t}\frac{\partial\mathcal L}{\partial \dot q_i}=\frac{\partial\mathcal L}{\partial q_i}$$

Of course to say why these equations (for $i=1,2...n$) are EoM there are needed little more studies in basics of classical mechanics :)

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