Orbital resonance

Orbital resonance is a resonance of two orbiting bodies exerting a regular, periodic gravitational effect on each other. Their orbital periods may be related by a ratio of two small integers. It is caused by the changing gravitational forces of bodies which go round each other. The stability of the Solar System was first investigated by Laplace, and there is still much that is not known about it.
As a satellite goes round a planet or two stars go round each other, the gravitational forces can change, sometimes hugely. This is partly because orbits are usually ellipses, not circular, and so the forces change accordingly. Also, the planets and stars are usually not spherical. They spin, and vary in their degree of oblateness. This also changes the forces on an orbiting body.
In particular, the forces may be unstable, so the smaller partner may change until the forces are stable (do not change with time). Satellites often end up with one face towards their planet, because that is the most stable position (tidal locking).
There are other stability effects. Gaps in Saturn's rings are caused by the particles shifting into more stable positions. In the rings of Saturn, the Cassini Division is a gap between the inner B Ring and the outer A Ring. It was cleared by a 2:1 resonance with the moon Mimas. Jupiter makes similar Kirkwood gaps in the asteroid belt.
There is a stability ratio for Neptune and Pluto: the 2:3 ratio means Pluto completes two orbits in the time it takes Neptune to complete three.
The area of mechanics which is used for these studies is called celestial mechanics.