Is it possible to harvest the energy from the movements of a satellite in orbit? I was thinking about how energy is harvested on Earth from movements of certain forces like wind and ocean currents. Could similar principles be applied in space?
Satellites are virtually in perpetual motion when orbiting the Earth. Is there kinetic energy that can be extracted from this orbital motion and harvested for use on Earth?
 A: Energy is 'harvested' from satellites all the time. Air drag converts the satellite's energy into heat in the atmosphere. This is, of course, not a perpetual motion machine. The orbit of the satellite will get closer to Earth over time, until the satellite crashes back down, as the Tiangong-1 did last week.
It should also be clear that you can't get more energy out of the satellite than you put in the rocket fuel in the first place. Even if you could throw a lasso around a satellite to take its energy, such a process would be incredibly inefficient. Similarly the energy harvested from wind is not because the wind is in "perpetual motion", instead it comes from the sun.
A: I do absolutely agree with @knzhou, but going to provide a couple of points here, more intuitive than scientific.
A satellite can spin for quite a long time since the kinetic energy of it doesn't decrease significantly (it doesn't mean that the energy doesn't decrease at all - it does, and for that reason a satellite will fall down eventually).
Now if we're going to harvest any energy with the aim of having a "profit", we would just "drain" it's kinetic energy, its speed will decrease at a greater rate, hence the satellite will fall noticeably sooner which would make the benefit of launching the satellite way less than the proposed benefit.

In other words, that's like trying to harvest an energy from a freely moving object. Harvesting would be pointless since the object would stop because of harvesting and you won't gain nothing.
I.e., you thought you have harvested the energy, but in fact your object has just passed all its energy to you and left with nothing, which is not what harvesting energy is supposed to be.

A side note regarding "perpetual motion" - it could be perpetual only in that ideal case when the energy is not lost. If you try to harvest energy from an object in "perpetual motion", the motion will no longer be "perpetual".
A: While I agree with other answers as to the physics of the problem, there is at least one practical area where harvesting energy from the orbital motion of satellites could be of practical utility: space debris removal system. As already mentioned, removing kinetic energy of orbiting body would result in this body falling back on Earth, but this would be the desired outcome for pieces of space garbage. One could imagine a system that removes the kinetic energy (for example, using tethers) from old defunct satellites causing their atmospheric reentry. Part of energy harvested is used to maintain the system in orbit and to maneuver it towards next piece of debris.
A: What knzhou and nicael said is absolutely true, you cannot extract from the satellite itself more energy of that you have put in it when you launch it in orbit.
Maybe you're interested in knowing how that energy could be extract, and I think that a space tether could be a proposal.
A: The energy conservation discussed by knzhou and nicael render the idea of harvesting energy from artificial satellites non useful.
However, energy is usefully and routinely harvested from natural satellites of the Sun in the manoeuvre of gravity assist aka "slingshot" which is used to accelerate spacecraft to speeds relative to Earth which would otherwise be extremely difficult using propellant alone. In particular, solar system escape, notably by the Voyager probes, has only ever been done using the energy harvested from natural Sun satellites to complement boost from propellants. 
The manouevre transfers some of the planet's kinetic energy and orbital angular momentum about the Sun to the spacecraft. Of course, the translational and spin kinetic energy of planets is utterly prodigious compared to any human-contemplated-artificial energy source, so the manouevre makes no noticeable difference to the assisting planet. Likewise, the acceleration of a spacecraft to $30{\rm km s^{-1}}$ relative to the Earth, which is needed for a Sun rendezvous to "deorbit" the spacecraft from its initially Earth-round-the-Sun orbit ($30{\rm km s^{-1}}$ being the Earth's orbital speed around the Sun), can only practically be done through a (usually Venus) gravity assist manouevre.
A neat simple analogy to get the idea across is given in this Physics SE answer by user mmesser314 here.
