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fluidistic
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Homework Statement
I must determine the conserved quantities+the equations of motion (of the trajectory in fact) of a particle over the surface of a cylinder.
Homework Equations
Lagrangian and Euler-Lagrange's equations.
The Attempt at a Solution
I've found the Lagrangian of the particle to be [itex]L=\frac{m}{2}(r^2\dot \phi ^2 + \dot z^2)[/itex].
Since the Lagrangian doesn't depend explicitly on [itex]\phi[/itex] nor [itex]z[/itex], the generalized momenta conjugate are conserved (I'm currently having under my eyes Goldstein's book, 1st edition, page 49).
So I can already answer this part of the problem, [itex]P_\phi=k_1[/itex] and [itex]P_z=k_2[/itex].
By intuition I know that the angular momentum is conserved and the speed of the particle along the z axis is constant.
I have a problem however with the Lagrange's equations.
For the generalized coordinate [itex]q=r[/itex] I have that [itex]\frac{\partial L}{\partial \dot r}=0[/itex] and [itex]\frac{\partial L}{\partial r}=m r \dot \phi ^2[/itex].
This gives me the first equation of motion, namely [itex]r\dot \phi ^2=0[/itex]. Since [itex]r\neq 0[/itex] (I'm dealing with a cylinder), [itex]\dot \phi =0[/itex].
Similarly, I get for [itex]\phi[/itex]: [itex]\underbrace{2 \dot r \dot \phi}_{=0} + r \ddot \phi =0 \Rightarrow \ddot \phi =0[/itex] which isn't a surprise since I already knew that [itex]\dot \phi=0[/itex].
I also get [itex]\ddot z=0[/itex].
So... the motion equations are [itex]\dot \phi =0[/itex] and [itex]\ddot z=0[/itex]?
I think I made a mistake. If I integrate them I get [itex]\phi = \text{ constant}[/itex] which is obviously wrong.
Hmm I'm confused about what I must do.
Edit: It seems that if I hadn't make any mistake, the motion equations should give me the information I already know: [itex]\dot \phi = constant[/itex] and [itex]\dot z = constant[/itex]. By the way I don't see my mistake for the Lagrange equation regarding [itex]\phi[/itex].
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