Goldstein Chapter 2: Derivation #2 Help

[SeReNiTy]

Can someone point me in the right direction with the derivation number 2 from Chapter 2 (3rd edition) of Goldstein?

 

[Kurdt]

Not everybody has the book unfortunately, but if you tell us what derivation it is then perhaps we can help.

 

[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.

 

[radou]

I've got Goldstein, and I'm lost on that one too. (As on most of them, btw. )

 

[SeReNiTy]

How would you define a potential with velocity dependent terms?

 

[Kurdt]

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:

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

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

$$\frac{\partial L}{\partial \dot{x}}= p_x$$

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 $$\mathbf{r}$$ rotated about any arbitrary unit vector $$\mathbf{\hat{a}}$$ by a small angle $$\theta$$.

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