A tower to geostationary orbit confused me quite a bit, until i realized there were two separate things to get confused about;then it confused me almost twice as much for a while ...

One is the gravitational potential itself, and the orbits thereof.

The other is the 24 hour rotation imposed on the problem by a rigid tower co-rotating with the earth.

Consider a demon who builds a tower to the heavens in Quito. He seizes you and drags you up it, only pausing to drop you off it evry nowanthen. Being demonic and bored, he retrieves your remains each time, reanimates them as necessary and proceeds, letting you go at a higher point each time. 23kilomiles up or so is the first time you dont move when he lets you go, but even before that you will have been in orbits with non grazing perigees (after splashing down in the Atlantic, making a crater in Africa and being eaten by sharks in the Indian Ocean, until perigee is outside the atmosphere, after which you merely expire from lack of air.)

As he passes geostationary point, your free orbit (the path you describe between demonic possessions) circularizes, the distance between foci of your orbital ellipse goes to zero and then reappears turned thru pi/2, and eccentricity increases again, until it crosses 1 at parabolic orbit and (earth) escape velocity point.

I have always found that imposing the daily co-rotating frame leads to all manner of complications such as hurricanes. And it isnt as tho Newtonian gravity doesn't have all manner of twisted arcana, all by itself, without considering how to work it out while chasing your tail evry day.

One of those arcana is the fifth conserved quantity.

Fizicists are a lazy bunch and the first thing they do when looking at a calculation are the symmetries they can use to finesse the calculations.

Gravity is a central (acts toward a single point), and an inverse square (varies inversely as the square of the distance) force. The central force bit already gives you a powerful symmetry: that of the rotational group in 3 dimensions, called the special orthogonal group or SO(3). You immediately know that the angular momentum, L, will be conserved, which is a 3-D vector, so that gives you 3 conserved quantities. You also know that time does no appear explicitly in the Newtonian equations. Therefore energy is also conserved, and that is another conserved quantity, and you have, immediately, 4 constants of the motion, quantities that are constant once you know the intial conditions (initial position and velocity of the orbiting mass.)

There are deep corners with inverse square forces like gravity; that magic minus 2 power ensures that orbits are closed (evry time the demon lets you go, you come back to his clutches after one orbital period, neglecting the orbits that end with a big kaboom,) but more; it is a peculiar symmetry that gives you one more conserved quantity related to the Laplace-Runge-Lenz vector, which is a litle vector that points from the central point of the force toward the other focus of the orbit and whose magnitude is proportional to the eccentricity. This is a result that exposes the deeper SO(4) and SO(3,1) symmetries and makes me feel all deep and tingly inside. probly coz i dont fully understand it.

There are more symmetries here, that of reflection and such, which brings in O(4) and other groups, but i have probly said too much already.

Tune in next time for another edition of the bloodshot eyeball of sidd.