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I am trying to express the first time derivatives $\dot{r}$ and $\dot{\phi}$ in a second order dynamic system in terms of $f(r,\phi), r$ and $\phi$ only: $$\ddot{r}-\frac{\dot{r}^2\tan^2{\phi}}{r}=f(r,\phi) \tan{\phi}$$ $$\ddot{r} \tan{\phi}+\dot{r}\dot{\phi}\sec^2{\phi}+\frac{\dot{r}^2 \tan{\phi}}{r}=f(r,\phi)$$

I timed $\tan{\phi}$ to the first equation and minus the second one to eliminate $\ddot{r}$ and it gave me a quite simple relation between the two first derivatives: $$\frac{\dot{r}^2\tan{\phi}}{r}+\dot{r}\dot{\phi}=f \cos{2\phi}$$

But I could not find another relation to let me figure out the exact expression of $\dot{r}$ and $\dot{\phi}$ in terms of $f(r,\phi), r$ and $\phi$. Is there a way to do so?

I'd really appreciate your help!

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