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fluidistic
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Homework Statement
Determine the possible trajectories of a particle into the following central potential: [itex]U(r)=U_0[/itex] for [itex]r< r_0[/itex] and [itex]U(r)=0[/itex] for [itex]r>r_0[/itex].
Homework Equations
Not sure. What I used: Lagrangian+Euler/Lagrange equations.
The Attempt at a Solution
I used polar coordinates but I'm not sure it's well justified. Since we're dealing with a central potential problem, isn't the motion restricted to a 2 dimensional plane? If so, then I think it's safe to use polar coordinates.
Ok so my Lagrangian is [itex]L=T-V=\frac{m(\dot \theta r^2 + \dot r ^2 )}{2}-U(r)[/itex]. I notice that [itex]\theta[/itex] is cyclic, thus the angular momentum is conserved.
Using Lagrange's equations for [itex]\theta[/itex], I reached the differential equation [itex]\dot r r = cte[/itex]. For [itex]r[/itex], I reach [itex]\ddot r - \dot \theta r =0[/itex].
I know I made an error, I reach the same motion equation regardless what U(r) thus r is.
Hmm... Since [itex]\theta[/itex] is cyclic, does this imply that [itex]\dot \theta =0[/itex]?
If so, then [itex]\ddot r =0[/itex]. But this is senseless since it would imply that [itex]\dot r[/itex] is also a constant, and thus that r is also a constant from my first use of Lagrange equations...
It might be a little too late to do physics for me now, but I'm willing to put the effort.
Any help will be appreciated. Thanks a lot!
Edit: I notice an error for the 1st Euler/Lagrange equation conclusion. I do reach that r is constant instead of [itex]r \dot r[/itex], which still doesn't make sense to me.
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