- #1
ASmc2
- 28
- 3
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...
Where Do Sections 3.45 and 3.46 in Goldstein's 3rd Edition Originate?
- Context: Graduate
- Thread starter ASmc2
- Start date
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Discussion Overview
The discussion revolves around the origins of equations 3.45 and 3.46 in Goldstein's 3rd edition textbook. Participants explore the derivation of these equations, focusing on the context of simple harmonic motion and the necessary mathematical steps involved.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant inquires about the origins of equations 3.45 and 3.46, indicating understanding of prior material.
- Another suggests starting with equation 3.34 and linearizing it about ##u = u_0## to derive the equations.
- A participant claims to derive an equation of the form x'' + kx = c, where c is a constant, and x represents u - u0.
- Another participant challenges this by stating that the correct form should be x'' + kx = 0, implying an error in the previous derivation.
- A participant expresses concern about terms involving u0^3 and u0 in their work.
- One participant asserts that the very first equation presented is incorrect, suggesting a differentiation error that affects the derivation of the third equation.
- A later reply acknowledges understanding after receiving clarification and indicates progress towards obtaining equation 3.46.
- Another participant states that equation 3.46 should follow directly from the previously derived equation, ##\ddot x + \beta^2 x = 0##.
- One participant expresses gratitude for assistance with equation 3.45.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the derivation process, as there are differing views on the correctness of the initial equations and the steps taken to arrive at the final forms.
Contextual Notes
There are unresolved mathematical steps and potential errors in differentiation that affect the derivation of the equations discussed. Specific assumptions about the forms of the equations and constants involved are not fully clarified.
- #2
voko
- 6,053
- 391
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.
- #3
ASmc2
- 28
- 3
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.
- #4
voko
- 6,053
- 391
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.
- #5
- #6
ASmc2
- 28
- 3
That u0^3 and u0 terms bother me, as you can see.
- #7
voko
- 6,053
- 391
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.
- #8
ASmc2
- 28
- 3
OK, I get that now, thanks. I got the correct equation now. Now to get (3.46).
- #9
voko
- 6,053
- 391
3.46 should follow from what you got immediately: ##\ddot x + \beta^2 x = 0 ##.
- #10
ASmc2
- 28
- 3
Yes. Thanks for the help with 3.45
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