Is a perfectly circular orbit possible?
Let's assume that there is a perfectly spherical planet and there is a moon which is also perfectly spherical. Let's assume that there is no atmospheric drag and no other gravitational pull. If the moon is put in a perfectly circular orbit around the planet somehow, would the moon eventually "fall" towards the planet and form an elliptical orbit or would it continue to follow the perfectly circular orbit?
Edit: What i actually wanted to ask is that would the gravity of the planet cause the moon to "fall" towards the planet or would the gravity allow the moon to continue its orbit without bending its path any further towards the planet. I know that no planet can be a true sphere or a cube due to the shapes of the particles.
Yes, "the gravity of the planet would cause the moon to "fall" towards the planet"; that's exactly what an orbit is!
One should note that the moon is not, strictly spoken, in an orbit around the planet; rather, both of them are orbiting the common center of mass. For small moons and large planets the difference is small, but for Pluto/Charon (and Charon's orbit is, userLTK says, fairly circular) it is significant: The center of the circle -- the *barycenter* of the Pluto/Charon system -- is outside Pluto.
Yes. If you ignore the tidal effect and relativity and any change in mass (planets radiate light and lose atmosphere and add space dust and meteors all the time, so mass isn't constant), then in a two body system with no outside effects, the orbit would remain perfectly circular. There would be no outside force to affect the circular orbit. A circular orbit is impossible because nothing can be that exact, but on a computer simulation you could set it up and it would remain circular.
For your scenario to work you'd need to give both the planet and moon infinite hardness, so they wouldn't bend at all and fixed mass and space would need to be completely empty of anything else. Needless to say that's impossible. But only in Newtonian gravity.
Relativity creates a very very tiny decay in orbits, in your system of a planet/moon that would be close to negligible but there would be a very tiny spiraling inward. The relativistic effect on an orbit was first noticed with Mercury's orbit around the sun (and Mercury isn't falling into the sun, it was noticed by other effects - but lets not get into that here).
Similarly, any loss in mass, gain in mass or orbital drag (because space is full of tiny particles, fast moving particles, photons and neutrinos, all of which cause a tiny but at least in simulation, calculatable drag), then the two body system would have an imperceptibly small spiral and not be a perfect circle. You could say in a sense that it becomes elliptical but it would be more like a constant very tiny force where, once it was elliptical, it could wonder back to more circular. Not all perturbations or drag on an orbit make that orbit more elliptical, it can work in either direction.
It's worth noting that "falling" or decaying towards the planet wouldn't "create" an elliptical orbit. A circle is an ellipse. You asked specifically about 2 body systems, where, ignoring tides, falling in or out would be more of a slow spiral. An ellipse isn't the result of a decaying or perturbed orbit. An ellipse is the baseline orbit. Perturbations and orbital decay happen on top of the ellipse (if that makes sense), they don't cause the ellipse.
In a 3 or more body system you get orbital perturbations on the orbits. Those often remain stable, they're just variations that mostly move back and forth. See Eccentricity variation and Apsidal Precession.
What if the moon is in tidal lock (as it indeed happens eventually, as we know)? And if it is in a geo- (or rather planeto-) stationary orbit, so that there is no rotational acceleration from the gravitational field of the planet either?
@PeterA.Schneider Maybe I'll update the answer with more details, but a perfect circle is impossible. A perfect tidal lock is also impossible, there will always be some wobble and some transfer of energy into heat. But you're right that a tidal lock like that is about as circular as it gets. Pluto-Charon is like that and very close to a circle.