Where Do Sections 3.45 and 3.46 in Goldstein's 3rd Edition Originate?

[ASmc2]

This is a question to those who have the 3rd edition at their disposal. Where do 3.45 and 3.46 come from? I understood everything he said before that...

 

[voko]

Start with equ. 3.34, an linearize it about $u = u_0$. You will get an equation for simple harmonic motion, from which those two equations will follow.

 

[ASmc2]

I get an equation of the form x''+kx=c. where c is some constant, '' gives the second derivative with respect to theta, and x is u-u0.

 

[voko]

You are on the right track, but you should have obtained $x'' + kx = 0$. You made a mistake somewhere. Without seeing your work, I can't say what it was.

 

[ASmc2]

My work is attached.

 

[ASmc2]

That u0^3 and u0 terms bother me, as you can see.

 

[voko]

The very first equation you have is incorrect. $$ {d \over du} V \left( \frac 1 u \right) \ne f(u) $$ Which is actually strange, because the third equation could only have been obtained if you had differentiated that correctly.

 

[ASmc2]

OK, I get that now, thanks. I got the correct equation now. Now to get (3.46).

 

[voko]

3.46 should follow from what you got immediately: $\ddot x + \beta^2 x = 0$.

 

[ASmc2]

Yes. Thanks for the help with 3.45