Goldstein Problem


by SeReNiTy
Tags: goldstein
SeReNiTy
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#1
Sep12-06, 01:05 AM
P: 171
Can someone point me in the right direction with the derivation number 2 from Chapter 2 (3rd edition) of Goldstein?
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Kurdt
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#2
Sep12-06, 04:45 AM
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Not everybody has the book unfortunately, but if you tell us what derivation it is then perhaps we can help.
SeReNiTy
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#3
Sep12-06, 06:06 AM
P: 171
The derivation is if the lagrangian contained velocity terms, derive what the conjugate momentum will be if the system is rotated by a angle in some direction.

radou
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#4
Sep12-06, 11:17 AM
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Goldstein Problem


I've got Goldstein, and I'm lost on that one too. (As on most of them, btw. )
SeReNiTy
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#5
Sep12-06, 08:41 PM
P: 171
How would you define a potential with velocity dependent terms?
Kurdt
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#6
Sep13-06, 07:23 AM
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Quote Quote by SeReNiTy
The derivation is if the lagrangian contained velocity terms, derive what the conjugate momentum will be if the system is rotated by a angle in some direction.
Well I hope you know the lagrangian is given by:

[tex] L=T-V = \frac{1}{2} m (\dot{x}+\dot{y}+\dot{z}) - V(x,y,z) [/tex]

In cartesian coordinates. The conjugate momentum for a particular coordinate is given by:

[tex] \frac{\partial L}{\partial \dot{x}}= p_x[/tex]

Now if the system was rotated by a particular angle how would you then modify the original expression to deal with a rotation. If you consider the x,y,z coordinates to be a vector [tex]\mathbf{r}[/tex] rotated about any arbitrary unit vector [tex]\mathbf{\hat{a}}[/tex] by a small angle [tex]\theta[/tex].

Now all you have to do is findout how the x,y,z coordinates are related to [tex]\mathbf{\hat{a}}[/tex] through vector [tex]\mathbf{r}[/tex] and plug them into the lagrangian and see what the turn up. In some books they set [tex]\mathbf{\hat{a}}[/tex] parallel to the z-axis for ease of computation and normally call it [tex]\mathbf{\hat{k}}[/tex] instead.

Like I say I don't have the textbook but I assume this is something to do with conservation of angular momentum?

Need any more pointers just post.


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