Rational ratio of frequencies leads to isolating integral of motion

[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.

$-\frac{\omega_{x}}{\omega_{y}}\cos^{-1}\left(\frac{y}{B}\right)+\cos^{-1}\left(\frac{x}{A}\right)=c$


This quantity $c$ is clearly another integral of motion. But- in general - this does not isolate the region where the motion takes place any further, because $\cos^{-1}z$ is a multiple-valued function. To see this more clearly, let us write

$x=Acos\left\{c+\frac{\omega_{x}}{\omega_{y}}\Big[Cos^{-1}\left(\frac{y}{B}\right)+2\pi n \Big]\right\}$

Where $Cos^{-1}z$ (with an uppercase C) denotes the principal value. For a given value of $y$ we will get an infinite number of $x$'s as we take $n=0, \pm 1, \pm 2, \dots$

Thus, in general, the curve will fill a region in the $(x,y)$ plane.

A special situation arises if $(\omega_{x}/\omega_{y})$ is a rational number. In that case, the curve closes on itself after a finite number of cycles. Then $c$ is also an isolating integral and we have three isolating integrals: $(E_{x}, E_{y}, c)$. The motion is confined to closed (one-dimensional) curve on the surface of the torus.

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?

 

[Greg Bernhardt]

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

 

[victorvmotti]

Thanks that you care about this question.

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.