Classical astronomy refers to research that does not involve statistics. Celestial mechanics, in the classical sense, can all be traced directly back to Newton’s Law of Gravitation. Hence, it’s sometimes called Newtonian astronomy.
The Two Body Problem
The original Bohr Model of the atom was a 2BP: one electron orbiting a much more massive nucleus. The equation for electrostatics governing the motion of that atom is identical to the Law of Gravity except for a constant of proportionality to account for the difference in scale.
Three Body Problem
A spacecraft (s/c) orbiting Earth is a 2BP. A s/c going to the moon is a Three Body Problem: the study of the motion of a tiny object under the gravitational influence of two much, much larger bodies like moons, planets, or stars.
Rotating Coordinates
In classical celestial mechanics, the motion of the small 3rd body is studied in a rotating coordinate system. The two large bodies are along the x-axis (the origin is their center of gravity), and the motion of the tiny 3rd body is plotted in these Cartesian coordinates.
The Free Return Orbit
The Apollo missions to the moon were on a figure-8-shaped orbit that was discovered in the mathematical investigation of the 3BP. Earth is inside one loop, and the Moon is inside the other; the loop orbits with Luna. In the rotating coordinate system, the orbit is a figure-8 along the x-axis.
A closed orbit in the 3BP is a zero-energy orbit. The Free Return Orbit does not require any input of energy by the s/c to remain on that trajectory. When the Apollo 13 module lost power, the Free Return Orbit returned them to Earth without the need for any propulsion.
The Trojan Asteroids
Any two large bodies constitute a 3BP. For example, the Sun-Jupiter system. There are two stability points in the 3BP – again, recognize that the Sun and Jupiter are fixed on the x-axis in this rotating coordinate system (it rotates with Jupiter) – at the apex of an equilateral triangle with sides equal to the Sun-Jupiter radius. The Trojan Asteroids orbit these stability points, following around with Jupiter’s orbit (but orbiting a fixed point in the rotating coordinates).
The 3BP in 3D
There are many closed loops in the full 3D Three Body Problem, each one a stable orbit. The Free Return Orbit is in the plane of the primary bodies. Other orbits are out of this plane.
Conclusion
All solutions in celestial mechanics can be easily verified by observation of astronomical phenomena. Since the equation for electrostatics is the same as for gravity, these solutions also exist in atoms.
Two nuclei and an electron are a 3BP. Many stable, zero-energy orbits at the subatomic level exist where electrons can persist in perpetual motion without expending any energy whatsoever.