non-repeating dynamic systems? real or not?

[charlie56]

Probably a duh question: If we define a mathematical object made up of two identical forever-rotating circles, everything perfectly regular (circular circles, axes central, rates of rotation unvarying, etc.) with the rates of rotation set in a non-integral ratio, such as 1 to pi, that is, the circles have a ratio of rotation that is a non-terminating, non-repeating fraction, will any particular state of that object ever recur--will two points that pass each other on a line through both axes ever be in that position again? I think that, because the ratio of rotation is non-integral, no state of the system is ever going to be repeated.

 

Of course, I am thinking in a plane--would the answer change in non-Euclidean spaces? What if the two circles are in different parts of a saddle-shaped 'plane', for instance? Are there other factors I haven't considered that could affect the outcome?

 

I am a non-mathematician--an editor--in a debate with another non-mathematician, who fancies himself a Nietzschean (Eternal Recurrence, bla bla). He says that, given enough time, any dynamic system with a limited number of variables must either stop (run out of steam, as it were) or repeat itself. For instance, given unlimited time and an unlimited energy source, the molecules of gas in a box will eventually return to some earlier state, and then repeat the same motions again, to infinity. I've argued that, since non-terminating, non-repeating fractions don't repeat, and since several such fractions seem to be fundamental constants in the universe (pi, e, etc.) the idea of a 'self-repeating cosmos', or of a self-repeating stochastic system the size of even a small sub-universe like ours, is implausible, and that the molecules might not ever return exactly to any previous state of energy and motion.

 

Of course, neither of us knows enough to resolve the dispute one way or the other.

 

My gratitude to anyone who takes the time to read this, and my profound thanks to anyone who undertakes an answer to any part of it. I realize some aspects of it are physics/thermodynamics related, and will put a related query into a physics-related forum.

 

Chuck

[overtone]

If time and space are fixed continuous, a priori infinities of the same "size" as the number line that includes the transcendentals etc, then you are correct - two physical setups moving in some unalterable deterministic fashion that involves exactly pi as some relative motion, will never repeat a position.

 

If time and space are a priori quantized, fundamentally discontinuous in some manner fixed in advance, no such relative motion is possible - all relations are rational, only a finite number of different relative positions exist, and in time all attainable states will be repeated.

 

If the continuity or quantization of time and space is of some currently undescribed nature, in which it is not fixed in advance say, then I don't know how to answer the question.

 

AFAIK nobody has established which of those three situations, or some other unimagined there, obtains.