- #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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SUMMARY
The discussion focuses on the derivation of equations 3.45 and 3.46 from Goldstein's 3rd Edition. The user initially misapplied the linearization of equation 3.34, leading to an incorrect form of the equation for simple harmonic motion. The correct form, x'' + kx = 0, is essential for deriving the subsequent equation, 3.46, which is expressed as \(\ddot x + \beta^2 x = 0\). The conversation emphasizes the importance of accurate differentiation in obtaining these equations.
PREREQUISITES- Understanding of linearization techniques in differential equations
- Familiarity with simple harmonic motion concepts
- Knowledge of second-order differential equations
- Proficiency in calculus, specifically differentiation
- Study the derivation of simple harmonic motion equations from linearized forms
- Learn about the implications of second-order differential equations in physics
- Explore the differentiation techniques relevant to potential energy functions
- Review examples of linearization in classical mechanics
Students of physics, particularly those studying classical mechanics, educators teaching differential equations, and anyone involved in the analysis of motion dynamics.
- #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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