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## Homework Statement

Hi everybody! I'm a bit stuck in this problem, hopefully someone can help me to make progress there:

A mass point ##m## is under the influence of a central force ##\vec{F} = - k \cdot \vec{x}## with ##x > 0##.

a) Determine the equation of motion ##r = r(\varphi)## for the angular momentum ##|\vec{L}| \neq 0##.

b) For which value of ##E## do we have a circle?

## Homework Equations

Lagrange equations, ##E = T+V##, ##V_{eff} = V(\vec{r}) + \frac{L^2}{2mr^2}##

## The Attempt at a Solution

Okay so first I determined the potential:

##V(\vec{r}) = \int_{|\vec{r}|}^{0} -kr dr = \frac{1}{2} k r^2##

Then I found the equations of motion

##\ddot{r} = r\dot{\varphi}^2 - \frac{k}{m}r## and

##\ddot{\varphi} = \frac{2}{r} \dot{r} \dot{\varphi}##

and I can deduce

##\frac{\partial V}{\partial t} = 0 \implies E = const.## and

##\frac{\partial L}{\partial \varphi} = 0 \implies \frac{d}{dt} \frac{\partial L}{\partial \dot{\varphi}} = 0 \implies \frac{\partial L}{\partial \dot{\varphi}} = mr^2 \dot{\varphi} = L = const.##

so I can rewrite ##\dot{r}## as

##\dot{r} = \frac{dr}{dt} = \frac{dr}{d\varphi} \dot{\varphi} = \frac{dr}{d\varphi} \frac{L}{mr^2}##

which I can substitute in my ##E##:

##E = \frac{1}{2} \frac{L^2}{mr^4} \bigg(\frac{dr}{d\varphi}\bigg)^2 + V_{eff}##

and after a few manipulations I get:

##\varphi = \pm \frac{L}{\sqrt{2m}} \int \frac{dr}{r^2 \sqrt{E - V_{eff}}}##

And that's where I get stuck. ##E## and ##L## are constants but surely ##V_{eff}## is not. In the Kepler problem we substitute with ##u = \frac{1}{r}## but if I am not wrong it is not working in the case of a simple oscillator. Any suggestion?

Thanks in advance for your answers, I appreciate it!

Julien.