How do you actually define an orbit?

I believe, Newtonian Mechanics describes an orbit as one object in free fall around another where projectile paths become elliptical. I think, Einstein describes an orbit as an object taking the shortest distance through curved space. And in Quantum Mechanics, orbits are quantized orbitals or states. Is there a definition for orbits, where all these characteristics are true?

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    $\begingroup$ In quantum mechanics we talk about "orbitals", not "orbits". The distinction may sound pedantic, but it is important because you should not be thinking about definite paths in the quantum mechanical realm. $\endgroup$ – dmckee Apr 4 '13 at 14:14

You could generalize to something like

The relative motion of two bodies under a mutual interaction with fixed angular momentum and a lower bound on the radial separation.

If you don't feel the need to examine asymptotically free interactions you could add an upper bound on the separation as well, but historically many people talk about parabolic and hyperbolic orbits.

You should distinguish the QM orbital from classical orbit as they are different beasts.

That said, trying to nail down one absolute, fixed and authoritative definition of a word that has been put to so many uses for so long is a little hopeless and not clearly useful. Just the longstanding use in free interactions should make it clear that people have not been welded to a single tight usage.

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    $\begingroup$ I don't think you need constant angular momentum for the motion of one object around another to be considered 'orbital'. For example, binary systems are in 'orbit' yet they lose angular momentum from gravitational wave emission. $\endgroup$ – user12345 Apr 4 '13 at 17:28
  • $\begingroup$ @user16307 Well, mostly. I would choose to treat that (and other processes like angular momentum transfer from tidal friction) as perturbations on a orbit which does have constant angular momentum. That means that I have to treat the end state of a gravitational radiation inspiral as a special case, but I'm OK with that. $\endgroup$ – dmckee Apr 4 '13 at 18:56
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    $\begingroup$ That's OK, but GWs are emitted throughout the entire life of a binary system, not just at the end. $\endgroup$ – user12345 Apr 5 '13 at 14:03
  • $\begingroup$ Yes, I know. In most orbits it's too small an effect to observe on the time scales of one time around the orbit so it's a perturbation. $\endgroup$ – dmckee Apr 5 '13 at 14:39

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