A Question on a Wikipedia article on Bertrand's theorem

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Kashmir
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Wikipedia while deriving Bertrands theorem writes after some steps :


(...For the orbits to be closed, ##β## must be a rational number. What's more, it must be the same rational number for all radii, since ##β## cannot change continuously.Using the definition of ##J## along with equation (1) we get##J^{\prime}\left(u_{0}\right)=\frac{2}{u_{0}}\left[\frac{m}{L^{2} u_{0}^{2}} f\left(\frac{1}{u_{0}}\right)\right]-\left[\frac{m}{L^{2} u_{0}^{2}} f\left(\frac{1}{u_{0}}\right)\right] \frac{1}{f\left(\frac{1}{u_{0}}\right)} \frac{d}{d u_{0}} f\left(\frac{1}{u_{0}}\right)=-2+\frac{u_{0}}{f\left(\frac{1}{v_{0}}\right)} \frac{d}{d u_{0}} f\left(\frac{1}{u_{0}}\right)=1-\beta^{2}##where ##J^{\prime}\left(u_{0}\right)=\left.\frac{d J}{d u}\right|_{u_{0}}##Since this must hold for any value of ##u_0## then##{\displaystyle {\frac {df}{dr}}=(\beta ^{2}-3){\frac {f}{r}}}## and hence##{\displaystyle f(r)=-{\frac {k}{r^{3-\beta ^{2}}}}.}##and then we find that##{\displaystyle J(u)={\frac {mk}{L^{2}}}u^{1-\beta ^{2}}.}##



However by a similar argument we can say that since ##β## is a constant then we can directly solve
##J^{\prime}(u)=1-\beta^{2}## and find that ##J=\left(1-β^{2}\right) u##) which is wrong.

What went wrong?



 
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