My question is during the initial deceleration, is that simply applied
opposite the current orbital velocity vector? And when the satellite
arrives at the new orbit, what direction is the correctional
acceleration/deceleration made? Is it the difference of the desired
tangential direction that it wants to go and the current direction it
is going? or simply opposite its current vector?
The two directions are the same. The impulses are done at the apoapsis and the periapsis of the transfer orbit, where the velocity is also tangential to the circular orbits (the larger one at apoapsis and the smaller at periapsis).
I understand that the point of the Hohmann transfer orbits is to use
as little velocity change (and thus fuel) as possible, but can an
orbital transfer be faster or slower if you're willing to burn more
fuel to affect the velocity change?
[snip]
Finally, if you had unlimited fuel, and time was more a consideration,
would you even bother with this process versus something more "point
and shoot, turn and burn"?
If you have enough delta-v and thrust, you can just "point and burn". You will be doing a hyperbolic transfer orbit that, in the limit of very large velocities, will tend to a straight line ($e \to \infty$).
No. Johannes Kepler published what is now known as his third law of planetary motion in 1619 (in his treatise Harmonices Mundi), but discovered it already on May 15, 1618. He simply related mean distance of a planet from the Sun to its mean angular motion, without a word about a mass, I think. He did write on gravity and mass (not the precise physical term) in a foreword to his earlier book Astronomia Nova.
Thanks to people (Rafael Gil Brand, Roger Ceragioli and R. H. van Gent) from H-ASTRO discussion forum I have the following update #1:
1, The original form of the third law (formulated for planets), freely traslated to English reads approximately:
"...it is absolutely certain and perfectly correct, that the ratio which exists between the periodic times of any two planets is precisely 3/2 of the ratio of the mean distances, i.e. of the spheres themselves, bearing in mind, however, that the arithmetic mean between both diameters of the elliptic orbit is slightly less than the longer diameter"
2, Although (as far as I know from my own experience with early observations of double stars by Galileo) it is virtually impossible to prove that an earlier observation/idea didn't exist, it seems that the first application of Kepler's Third Law to the Jovian satellite system, is found in Newton's Philosophiae Naturalis Principia Mathematica (2nd ed. of 1713), lib. III, prop. 8, resulting in 1/1033 solar mass.
It is possible that Riccioli had something about the topic in one of his monumental treatises published around the middle of the 17th century.
Update #2
Riccioli seem to discuss relation between elongation of Galilean satellites of Jupiter and their orbital periods both in his Almagestum Novum and Astronomia Reformata, and cites Vendelinus (Godefroy_Wendelin). The Wikipedia entry for him states:
"In 1643 he recognized that Kepler's third law applied to the satellites of Jupiter."
without further details.
Update #3 - Final answer
I repost here the final answer by Christopher Linton from H-ASTRO:
"Kepler, in the Epitome of Copernican Astronomy (1618-1621), did apply his third law to the Jovian satellites (in Art. 553). He got the data from Simon Mayr's World of Jupiter (Mundus Jovialis, 1614). He establishes that $T^2/a^3$ is roughly constant and concludes that the physical mechanism which causes the planets to move as they do is the same as that which causes the Jovian satellites to rotate around Jupiter."
Best Answer
Use perturbation theory. Increase and decrease some moon speed a little and see that the forces from the other moons tend to counteract the perturbations. The same technique used in electronics to stabilize frequency oscillators is phased-lock loop PLL.