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Solution to the radial energy equation

  1. Aug 4, 2014 #1
    1. The problem statement, all variables and given/known data
    Find the polar equation of the orbit of an isotropic harmonic oscillator by solving the differential equation [itex]\frac{J^{2}}{2M}[/itex]([itex]\frac{dv}{dθ}[/itex])[itex]^{2}[/itex][itex]\frac{1}{v}[/itex] + [itex]\frac{J^{2}}{2M}[/itex]v + V(r) = E. And verify that it is an ellipse with centre at the origin.

    2. Relevant equations
    V(r) = 0.5kr2

    v = 1/r2

    3. The attempt at a solution

    My attempt at the solution is in the attached photo

    I have got the solution of r2[esinθ + 1] = l

    where l = J2/(Em2) and e2 = 1- (kl)/(Em)

    When I plot this i do get an ellipse but it is not centred on the origin, i'm also having trouble verifying that it is the equation for an ellipse by substituting r2 = x2 + y2, and rsinθ = y, rcosθ = x. Im not sure if my solution is correct? Thanks.
     

    Attached Files:

    Last edited: Aug 4, 2014
  2. jcsd
  3. Aug 4, 2014 #2

    TSny

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    Homework Helper
    Gold Member

    In transforming from the variable u to the variable v, it looks to me that there is an error. In particular, the first energy equation in the right column of your hand-written notes has the wrong numerical coefficient of 2 in the denominator of the first term.

    Also, later when you multiplied through by 2/(EM), you dropped a factor of (1/M) in one of the terms. A dimensional analysis of the terms might be helpful.
     
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