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jamie.j1989
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Homework Statement
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.
Homework Equations
V(r) = 0.5kr2
v = 1/r2
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. I am not sure if my solution is correct? Thanks.
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