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Path Equation for 2D weakly-anisotropic harmonic oscillator

  1. Jul 26, 2010 #1
    1. The problem statement, all variables and given/known data
    [tex]\omega_{x}[/tex] = [tex]\omega[/tex]

    [tex]\omega_{y}[/tex] = [tex]\omega[/tex] + [tex]\epsilon[/tex]

    where 0 < [tex]\epsilon[/tex]<<[tex]\omega[/tex]

    Question: Find the path equation.

    2. Relevant equations

    I started with the 2D equations:

    x(t) = A[tex]_{x}[/tex]cos([tex]\omega_{x}[/tex]t + [tex]\phi_{x}[/tex])
    y(t) = A[tex]_{y}[/tex]cos([tex]\omega_{y}[/tex]t + [tex]\phi_{y}[/tex])

    3. The attempt at a solution

    by inverting x(t) to get t(x), I then substituted the result into y(t). The result is as below:

    y(x) = A[tex]_{y}[/tex]cos[([tex]\omega_{x}[/tex]/[tex]\omega_{y}[/tex])cos[tex]^{-1}[/tex](x/A[tex]_{x}[/tex]) - ([tex]\omega_{x}[/tex]/[tex]\omega_{y}[/tex])[tex]\phi_{x}[/tex] + [tex]\phi_{y}[/tex]]

    I guess it becomes more of a mathematical problem. How do I simplify this equation, hopefully to find something familiar? An idea I had was to use double-angle trig formulas but I am not sure how it would help.

    I personally think this is quite challenging. I have been thinking about it for days now. I tried all the trig identities I know off to manipulate the equation but I can't seem to get it. Unless I am missing something.

    Thanks in advance.

    PS: A^x is actually A(subscript)x and similarly, A^y is actually A(subscript)y. I don't know what is wrong with the formatting.
     
    Last edited: Jul 26, 2010
  2. jcsd
  3. Jul 26, 2010 #2
    maybe I should define what the symbols are:

    [tex]\omega_{x}[/tex] is the angular frequency in the x-axis
    [tex]\omega_{y}[/tex] is the angular frequency in the y-axis
     
  4. Dec 11, 2010 #3
    I'd try to substitute the [tex]\omega[/tex] you have and then try to expand it by small parameter [tex]\epsilon[/tex]...
     
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