I am going to speak of orbits in terms of the 2 body problem. The fundamental assumption is that the 2 bodies in question have masses that are "far greater" than any other masses "nearby". This is a simplifying assumption for the sake of example.

There are 3 possible interactions that can take place in the 2 body problem.
1) The bodies will orbit each other in elliptical paths.
2) The bodies will move past each other in parabola shaped paths.
3) The bodies will move past each other in hyperbola shaped paths.

Newtonian mechanics: There is a critical speed which determines which path is taken. This is the escape velocity (it should be named escape speed, since direction is irrelevant). If the bodies' relative speed (the speed of one body in the other body's reference frame) is less than their escape velocity, they will have closed, stable orbits about each other. If they are moving at escape velocity they will swing past each other once in a parabola. If they are moving above escape velocity, they will swing past each other once in a hyperbola.

Local potential: I don't know. I'm sure there is an answer, but I don't have it.

Path of least action: For a closed, stable orbit, there is some path from A to A over some period of time for which the change in energy is zero. This path is the orbit.

General Relativity: Space-time is curved by the presence of mass. The curvature of space-time contains certain paths, called geodesics, which are closed loops. Imagine contour lines on a topographical map, except in 3-D space. If a body is moving along one of these lines, it will continue to move along the line without deviation unless acted on by some outside force. (Echos of Newton's First).

Quote Originally Posted by Renton View Post
Thanks a lot for explaining this, I find it extremely interesting.
The pleasure is all mine. With all I'm learning from reading and re-reading your many informative posts, it is nice to be able to give something back. The thanks are heartily returned.

Quote Originally Posted by Renton View Post
So according to Newton, in order for Earth to orbit the Sun, it must have been in a state of accelerating in a direction not toward the Sun, i.e. Newton's cannonball. But relativity explains this better?
Short answer: not accelerating, moving is enough. The Earth had some momentum which was not toward the sun, which, in the lack of friction or some other dissipating force, is still present.

Relativity explains it differently, but "better" is subjective. From a strictly physics viewpoint, General Relativity is the most accurate model. From a "put an SUV on Mars" viewpoint, Newtonian mechanics is accurate enough without being needlessly complex. Except for the orbit of Mercury, Newtonian mechanics is adequate to describe the solar system.

Long answer:
The rotations in the solar system have been present from its earliest formation. When the sun's prior incarnation (our sun has gone supernova at least once before) was first formed in the collapse of a stellar nebula, there was rotation. The atoms in the nebula were not falling straight toward the gravitational center. The nebula collapsed in a swirling spiral, and created a spinning star. This star had no planets, as the nebula which formed it contained mostly Hydrogen and some Helium.

Eventually, this star exploded in a supernova and spewed its atoms back out into a new stellar nebula. This new nebula had exactly the same total rotation as the star that created it, which had exactly the same total rotation as the nebula from which it formed. However, now the atoms are different. There is an abundance of "heavy metals" that were created in the supernova, not to mention the Carbon, Nitrogen, and Oxygen that were created in vast quantities by "normal" fusion before the supernova.

Another collapse, another spinning star, we'll call this star the sun. Now, there are heavy elements causing local areas of high density, which act as minor gravitational centers. The collapse is less uniform, and planets, moons, comets, etc. are formed. Still, the total rotation in the system hasn't changed for billions of years, this is called the law of conservation of angular momentum.

Due to the law of conservation of angular momentum (one of the strongest conservation laws known), the solar system today still bears that rotation. Some of it is in the rotating of the sun, planets and moons. Some of that rotation is in the orbits of these bodies.