Rational ratio of frequencies leads to isolating integral of motion

1. Aug 9, 2014

victorvmotti

Hello All,

Padmanabhan's discussion of dynamics mentions that in general the two dimensional harmonic oscillator fills the surface of a two torus.

He further notes that there will be an extra isolating integral of motion provided that the ratio of frequencies is a rational number.

This last part is not still clear to me.

Can someone please explain why a rational ratio of frequencies make a candidate integral of motion single valued and therefore the motion takes place on a closed (one dimensional) curve on the surface of the two torus?

Last edited: Aug 9, 2014
2. Aug 13, 2014

Greg Bernhardt

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

3. Aug 13, 2014

victorvmotti

Well I got the answer later from another source.

My confusion was that in the case of a rational ratio, even though periodic, we have still multiple (finitely) values and not a single valued variable.

Turns out that like the nth root of unity in complex plane we can define a single valued function over a multiple (finitely) valued variable. And therefore in the case of a rational ratio we have a new isolating integral of motion which limits the dimension of phase space to just one instead of four.

Clearly in the case of the irrational ration you cannot have a periodic valued variable and therefore the phase space dynamic covers the whole surface of the two torus.