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Hi everyone
I have a cone, upside down and a mass m in it, also a homogenous gravitiy field. I found the EulerLagrange equation already, which led to 2 equations of motion. Now I want to find a stationary path ( acceleration=0)

The solution for the 2 ELE looks like:
[tex] \ddot r(1+cot^{2}( \alpha))r \dot \phi^{2}+g*cot( \alpha)=0[/tex]
and
[tex] r \ddot \phi +2 \dot r \dot \phi =0[/tex]
I shall only find ONE path. My guess would be to take the second equation in order to get a linear differential equation. If the acceleration equals 0, it simplifies to:
[tex] \dot r \dot \phi =0[/tex]
But I'm a bit confused on how to solve this now? Any hints? Or did I do some mistakes?
Thanks for your help
edit: If that's correct thus far, it's just a constant I guess, but do I know sth about it?
2nd edit: if that's right phi is constant so the mass is just falling downwards?!
Homework Statement
I have a cone, upside down and a mass m in it, also a homogenous gravitiy field. I found the EulerLagrange equation already, which led to 2 equations of motion. Now I want to find a stationary path ( acceleration=0)
Homework Equations

The Attempt at a Solution
The solution for the 2 ELE looks like:
[tex] \ddot r(1+cot^{2}( \alpha))r \dot \phi^{2}+g*cot( \alpha)=0[/tex]
and
[tex] r \ddot \phi +2 \dot r \dot \phi =0[/tex]
I shall only find ONE path. My guess would be to take the second equation in order to get a linear differential equation. If the acceleration equals 0, it simplifies to:
[tex] \dot r \dot \phi =0[/tex]
But I'm a bit confused on how to solve this now? Any hints? Or did I do some mistakes?
Thanks for your help
edit: If that's correct thus far, it's just a constant I guess, but do I know sth about it?
2nd edit: if that's right phi is constant so the mass is just falling downwards?!
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