Rational ratio of frequencies leads to isolating integral of motion

In summary, Padmanabhan discusses that in general the two dimensional harmonic oscillator fills the surface of a two torus. If the ratio of frequencies is a rational number, then there is an extra isolating integral of motion provided that the curve closes on itself after a finite number of cycles.
  • #1
victorvmotti
155
5
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.

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


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

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

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

Thus, in general, the curve will fill a region in the [itex](x,y)[/itex] plane.

A special situation arises if [itex](\omega_{x}/\omega_{y})[/itex] is a rational number. In that case, the curve closes on itself after a finite number of cycles. Then [itex]c[/itex] is also an isolating integral and we have three isolating integrals: [itex](E_{x}, E_{y}, c)[/itex]. 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?
 
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  • #2
I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?
 
  • #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.
 

FAQ: Rational ratio of frequencies leads to isolating integral of motion

1. What is the rational ratio of frequencies in the context of motion?

The rational ratio of frequencies refers to the relationship between two frequencies that can be expressed as a ratio of two integers. In the context of motion, this ratio is often used to describe the relationship between the frequencies of two objects moving in a periodic motion, such as two planets orbiting a star.

2. How does the rational ratio of frequencies lead to isolating an integral of motion?

The rational ratio of frequencies is useful in isolating an integral of motion because it allows us to identify a specific quantity that remains constant during the motion of the two objects. This integral of motion is often related to the conservation of energy or angular momentum.

3. What is an integral of motion?

An integral of motion is a quantity that remains constant during the motion of an object or system. It is often related to a fundamental physical principle, such as the conservation of energy, momentum, or angular momentum. In the context of the rational ratio of frequencies, the integral of motion is related to the conservation of energy or angular momentum.

4. How is the rational ratio of frequencies used in studying periodic motion?

The rational ratio of frequencies is a useful tool in studying periodic motion because it allows us to identify the relationship between the frequencies of two objects and to isolate an integral of motion. This helps us to better understand the behavior and dynamics of the system, and to make predictions about its future motion.

5. What are some real-world examples of the rational ratio of frequencies in action?

One example of the rational ratio of frequencies in action is the relationship between the moon's orbital frequency and the rotation frequency of the earth. Another example is the oscillation of a pendulum, where the frequency of oscillation is related to the length of the pendulum and the acceleration due to gravity. These examples demonstrate the use of the rational ratio of frequencies to isolate an integral of motion and understand the behavior of a system.

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