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Anisotropic Oscillator Consider a two-dimensional anisotropic oscillatoris rational (that is, ws/wy = p/q4.Prove that if the ratio of the frequencies w and wywhere p and q are integers with gcd(p, q) = 1, where gcd stands for greatest commondivisor) then the motion is periodic. Determine the period.P

Question
Anisotropic Oscillator Consider a two-dimensional anisotropic oscillator
is rational (that is, ws/wy = p/q
4.
Prove that if the ratio of the frequencies w and wy
where p and q are integers with gcd(p, q) = 1, where gcd stands for greatest common
divisor) then the motion is periodic. Determine the period.
P
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Anisotropic Oscillator Consider a two-dimensional anisotropic oscillator is rational (that is, ws/wy = p/q 4. Prove that if the ratio of the frequencies w and wy where p and q are integers with gcd(p, q) = 1, where gcd stands for greatest common divisor) then the motion is periodic. Determine the period. P

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Step 1

Assuming the ratio of the frequencies to be rational as mentioned in the problem.

Let τ be the smallest time period after which the motion along the x-direction is repeated.

This is expressed as,

2тр
1
о
х
Т%3
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2тр 1 о х Т%3

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Step 2

Here p is an integer, ωx is the frequency of oscillation of the anisotropic oscillator along the x axis.

From the relation between ωx and ωy given in the problem, τ can also be represented in terms of the frequency ωy of the anisotropic oscillator along y axis

2πα
τ=
ω.
о,
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2πα τ= ω. о,

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Step 3

The solution for the anisotropic oscillator along ...

x(+r) Acos(t+r
-Acos0+2p
- Acos at
= x(t)
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x(+r) Acos(t+r -Acos0+2p - Acos at = x(t)

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