Can someone point me in the right direction with the derivation number 2 from Chapter 2 (3rd edition) of Goldstein?
Not everybody has the book unfortunately, but if you tell us what derivation it is then perhaps we can help.
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.