Impossibility of perpetual motion as a mathematical theorem Can the impossibility of perpetual motion be expressed as a mathematical theorem?
 A: Yes.  If by 'perpetual motion' you mean a machine which runs for ever which produces some nonzero output power (that is, you can extract energy from it at some nonzero rate), then to prove it impossible you merely need to prove energy conservation, and this follows as a result of Noether's theorem as a result of the invariance of the laws of physics under time translation.
In other words, if the laws of physics remain the same over time, then such a machine can not exist.
Such machines are known as perpetual motion machines of the first kind: there are other kinds.
A perpetual motion machine of the second kind is a machine which converts heat into mechanical work, needing only one reservoir to do so.  An example of such a machine might be some device which, if you place it a large hot reservoir, will endlessly (or until the hot reservoir is exhausted) do mechanical work, without needing any cold reservoir.  These machines can be shown to violate the second law of thermodynamics and so can't exist.
Note that the first law of thermodynamics is a version of the law of energy conservation for thermodynamic systems, so you could say that perpetual motion machines of the first kind violate the first law of thermodynamics, while those of the second kind violate the second law.
A perpetual motion machine of the third kind is a different thing: these are machines which, once started, will run for ever, with no requirement to be able to extract energy.  These can't exist because, really of dissipative effects.  However in a sense certain special kinds of them can exist.  If you think of a hydrogen atom in its ground state as a sort of simple machine, then it will run for ever once it is set going (if you assume that protons and electrons don't decay).
