Rational ratio of frequencies leads to isolating integral of motion

  1. Aug 9, 2014 #1
    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. jcsd
  3. Aug 13, 2014 #2

    Greg Bernhardt

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    Staff: Admin

    I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
  4. Aug 13, 2014 #3
    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.
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