How does physics deal with perpetual motion in orbits? This is a bit a bit of a weird question. Simply put, if a body orbits something such as a black hole there is no internal issue (that I know of) that would cause it to eventually stop (by internal I mean that an external force can't be applied as an example). I'm assuming the reason this 'perpetual motion machine' fails is one of the following:

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*The laws of thermodynamics do factor in external issues (because those laws are what prevents perpetual motion)


*This would only work if every object could maintain their gravity (proton decay, hawking radiation etc would pretty much decay the system) and the laws of physics say that they cannot.
To reiterate because see this a lot: I'm not asking if this is a perpetual motion machine, I'm asking why it isn't.
 A: If a body orbits something else, then there is gravitational waves (GW) emission and the orbit shrinks (eventually the two bodies will merge).
Neglecting GW emission (assuming Newtonian dynamics is the correct description), the motion goes on forever, but this is not problematic since any kind of friction is neglected. So the system is no more or less problematic than an oscillating mass on a spring, as long as there is no friction and the spring does not heat up.
Hence, such a system is  a "perpetual motion machine of the third kind", defined as one that completely eliminates friction and other dissipative forces, to maintain motion forever due to its mass inertia. Clearly, it is impossible to build such a machine (in your two-body example, it is only possible on paper because you are neglecting GW emission or friction with the low-density space medium).
Most importantly, a machine is useful if you can extract work from it. Extracting work from an orbit it's possible, but again.. the orbit will "shrink" (for example, extracting work from an orbit is used to accelerate spacecrafts: to increase speed, the spacecraft flies with the movement of the planet, acquiring some of the planet's orbital energy in the process).
