A Doubt in a step while deriving Bertrand theorem

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The discussion revolves around the derivation of Bertrand's Theorem as presented in Goldstein's 2nd edition, specifically addressing the use of Fourier expansion in the context of the orbit equation under a conservative central force. The author questions the rationale behind employing only cosine terms in the Fourier expansion of the deviation from circularity, ##x##, particularly in relation to the evenness of the right-hand side of equation A-10. It is suggested that the evenness of the solution may stem from the nature of the orbit equation, where if ##u(x)## is a solution, then ##u(-x)## is also a solution. The discussion highlights the importance of understanding the conditions under which the right-hand side of A-10 behaves as an even function of ##\theta##. Clarity on these conditions is essential for justifying the choice of Fourier terms in the derivation.
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Goldstein 2nd ed.

In its Appendix is given the derivation of Bertrands Theorem.
Screenshot_20211029_103327.jpg
Here ##x=u-u_0## is the deviation from circularity and ##J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)##

If the R.H.S of A-10 was zero, the solution was then ##a \cos(β\theta)##. However if there are terms on the RHS as given in equation A-10 the author writes the solution as a Fourier sum involving only cosine terms .
Now how does the author know that we should use a Fourier expansion of ##x## using only cosine terms with argument ##β\theta##?
 
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Do you have a condition that RHS of A-10 is even function of ##\theta## ?
 
anuttarasammyak said:
Do you have a condition that RHS of A-10 is even function of ##\theta## ?
We started from the orbit equation under a conservative central force written as
##
\frac{d^{2} u}{d \theta^{2}}+u=J(u)
##
where
##
J(u)=-\frac{m}{l^{2}} \frac{d}{d u} V\left(\frac{1}{u}\right)=-\frac{m}{l^{2} u^{2}} f\left(\frac{1}{u}\right)
## which i believe if ##u(x)##is its solution then ##u(-x)## is as well.
So maybe the eveness of solution comes from here. I'm not sure.
 
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