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fluidistic
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Homework Statement
Exactly the same problem as https://www.physicsforums.com/showthread.php?p=3335113#post3335113 but instead of a cylinder, the surface is a cone.
Homework Equations
Same as previous thread.
The Attempt at a Solution
I used cylindrical coordinates [itex](r, \phi , z)[/itex].
By intuition I know the angular momentum with respect to the z-axis (axis of symmetry of the cone) must be conserved, thus the Lagrangian must not contain [itex]\phi[/itex].
I reached that the Lagrangian [itex]L=\frac{m}{2}(\dot r ^2 + r^2 \dot \phi ^2 + \dot z ^2)[/itex].
Now, I believe I must express r in function of z. I notice that if [itex]\theta[/itex] is the angle worth half the angle of the vertex of the cone, then [itex]\tan \theta = constant = \frac{r}{z} \Rightarrow r=k_1z[/itex].
So now I have 2 cyclic coordinates and it means that the momentum [itex]P_r[/itex] is a constant. Since r is directly related to z, this also means that the z-component of the linear momentum is constant... Well I think so.
Calculating Lagrange's equations, I reach as equation of motion: [itex]\ddot z (K+1) - k_2 z \dot \phi ^2=0[/itex]. This can be written as [itex]K_3 \ddot z + K_2 z \dot \phi =0[/itex]. Where [itex]K_3 >1[/itex].
Am I right?
If so, how can I solve it?! What method would do the job?