Could Earth theoretically have another moon, beside the existing one, that would have a stable orbit? If it is possible, could Newtonian physics describe where it would be (i.e its position relative to Earth and the Moon)?
Certainly the Earth could have ended up with additional moons, and Newtonian physics can say some things about the kinds of orbits they might have, but there is a range of possibilities.
First of all, stable orbits in multi-object systems tend to be fairly circular, just because objects in highly elliptical orbits will cross the orbits of other bodies and eventually collide, or be flung out of the system, or end up in a non-interfering orbit. (But, for instance, comets have highly elliptical orbits around the sun that can last for many centuries. The thing is, they tend to get perturbed into different orbits on longer time scales.)
Another general pattern, within the category of circular-ish orbits, is the phenomenon of resonance. Sometimes moons will end up in stable patterns where, for instance, the inner moon orbits twice every time the outer one orbits once. Around Jupiter, for every four orbits of Io, Europa orbits twice, and Ganymede once. So we might see that pattern. (But there is no guarantee. The different planets don't have resonances with each other, for instance, and those orbits operate under the same rules. But Pluto, no longer a planet, does have a 3:2 resonance with Neptune.)
Also, objects in circular orbits with very different radii don't effect each other as much as they would if the orbits were similar. And the moon's orbit allows a lot of room. If you can picture a vinyl LP with the earth filling up the hole in the center, then the edge of the record is about the moon's orbit. So a much closer moon is plausible.
But these are all just general thoughts. It turns out there is a recent study on this: "Moon-packing around an Earth-mass Planet" by Satyal, Quarles, and Rosario-Franco. The authors find that the earth could, in principle, have more than ten moons the size of the asteroid Ceres, six the size of Pluto (smaller than our moon), and four the size of the moon. The article linked didn't say anything about the kinds of possible orbits, but I am confident they would be approximately circular and not too close together.