Can other terrestrial moons be (or have been) stable?
Regarding other possible moons around the Earth, there should not be any other moons closer to Earth than the Moon is today. According to the currently believed formation scenario, the Moon formed by accretion of material ejected in a giant impact between the Earth and a protoplanetesimal on an orbit much closer to the Earth than the current one. It then evolved by tidal interaction with the Earth to the current orbit. Any moon whose orbit was between the original orbit of the Moon and today's one should have been removed by collisions with our satellite.
Moons on orbits outside the lunar orbit should be unstable due to solar or lunar perturbations.
Thank you for your question at "Ask an Astronomer". Your statement about why the Earth has only one Moon is correct. Let's just summarize it and look at it in further detail. The problem about the possible existence of other satellites of the Earth can be divided in three parts:
1) Satellites whose semimajor axis is less than that of the Moon today.
As you said, it is currently believed that the Moon formed by accretion of terrestrial material expelled from a giant impact much closer to the Earth than its current position, and then its orbit progressively expanded due to the effect of tides. Any satellites in an orbit between the original value of the semimajor axis of the Moon and the current value would have collided with the Moon and been removed from the system. An interesting article on this subject was written by Runcorn (1982, Primeval displacement of the lunar pole. Physics of the Earth and Planetary interiors 24, pp. 205-217). In this article the athor suggested that the formation of the sea basins on the Moon could have been caused by the impact of terrestrial satellites. The formation of the basin would have changed the moment of inertia of the Moon, and so also the spin axis. That would explain why the major "Maria" are roughly aligned in three big circles.
2) Satellites further away than the current position of the Moon.
You are again right. The Moon itself is already very distant from the Earth (its current position is at about 55% of the Hill sphere radius, that is the distance over which the gravitational influence of the Earth is stronger that tidal perturbations from the Sun). Satellites that are further away would be strongly perturbed not only by the Moon, but by the Sun itself, and most of them would either escape from the Hill sphere of the Earth or collide with the Moon or the Earth. So, no other external satellite is expected. It remains only one last possibility, which is:
3) Satellites in orbits around one of the two stable Lagrangian points (L4 and L5).
As you may know, in the three body problem there are two points that are stable, L4, and L5, which are points on the same orbit of the second mass (the Moon, in this case), but 60 degrees away from the satellites (their position forms an equatorial triangle with the first and second masses). So, an ideal place to look for other terrestrial satelites could be one of this two points, in theory. The problem is that in the real solar system there are other perturbations besides those of the Earth and the Moon, and orbits that are stable in the three-body problem may not be such in the real system. A study of the stability of the two lunar lagrangian points has been carried out by Jorba A. (2000. A numerical study on the existence of stable motions near the triangular points of the real Earth-Moon system. A dynamical systems approach to the existence of Trojan motions. Astronomy and Astrophysics 364, pp327-338). His conclusions are that tadpole orbits around L4 and L5 are unstable, but there are trajectories that remain close to the equilateral points for at least 1000 years. So, in principle, it is possible that temporalily captured small satellites (~100 m in diameter or less) or dust is currently present in one of these orbits.
Thank you again for your question, I hope that this answers or further clarifies what you stated.
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