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I have a small question that bothers me. Consider a test particle in the Earths gravitational field on circullar obrit. Speciffic effective potential is

[tex]

\begin{equation}

\widetilde{V}_{\rm eff(1)} = \frac{1}{2} \frac{l^2}{m^2} \frac{1}{r^2} - \frac{GM}{r},

\nonumber

\end{equation}

[/tex]

where

[tex]

\begin{equation}

l = m r^2 \Omega = {\rm const.}, \nonumber

\end{equation}

[/tex]

because [tex]$\theta$[/tex] is cyclic coordinate. But we have partice in circular orbit, so [tex]$\Omega = $[/tex]const. too, and we can write

[tex]

\begin{equation}

\widetilde{V}_{\rm eff(2)} = \frac{1}{2} \Omega^2 r^2 - \frac{GM}{r}.\nonumber

\end{equation}

[/tex]

Now we have correct form:

[tex]

\begin{equation}

\frac{ \partial^2 \widetilde{V}_{\rm eff(1)}}{\partial r^2} = \frac{3 l^2}{m^2 r^4} - \frac{GM}{r} = 3 \Omega^2 - \frac{2GM}{r^3}.\nonumber

\end{equation}

[/tex]

and incoorect form

[tex]

\begin{equation}

\frac{ \partial^2 \widetilde{V}_{\rm eff(2)}}{\partial r^2} = \Omega^2 - \frac{2GM}{r^3}.\nonumber

\end{equation}

[/tex]

Why is there a difference between [tex]

$ \partial^2 \widetilde{V}_{\rm eff(1)} / \partial r^2, $ [/tex] and [tex] $\partial^2 \widetilde{V}_{\rm eff(2)} / \partial r^2, $

[/tex]?

Thanks for any replies, have a nice day with physics :-)

Martin