Solution to the radial energy equation

[jamie.j1989]

Homework Statement

Find the polar equation of the orbit of an isotropic harmonic oscillator by solving the differential equation $\frac{J^{2}}{2M}$($\frac{dv}{dθ}$)$^{2}$$\frac{1}{v}$ + $\frac{J^{2}}{2M}$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.

 

[TSny]

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.