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Homework Help: Orbital Motion Confusion

  1. Nov 25, 2013 #1
    Hi, I stumbled upon this while working on a problem on my physics homework. I still want to solve the problem myself if possible though so I won't post it here, instead, I'll post what is confusing me.

    Consider orbital motion with potential U(r), where U(r) is any arbitrary function of r.
    I was able to show that the quantity [itex]L=mvr[/itex] is conserved and I will call it L. Thus:

    We know that the system has a total energy that is constant:
    [itex]U(r)=E-\frac{m}{2} \frac{L^2}{(mr)^2}[/itex]

    This shows that potential is only dependent on radius. Everything else is a constant. Furthermore, it shows that potential as a function of radius is ALWAYS equal to the same thing... This simply cannot be true... Where am I going wrong?

    The problem that I'm working on gives me a function [itex]r(\theta)[/itex] and asks "What central force is responsible for this motion".

    Using the method above... I'm finding that F(r) is the same thing no matter what [itex]r(\theta)[/itex] is... (By taking the negative derivative of U(r) with respect to r.)
  2. jcsd
  3. Nov 25, 2013 #2
    The angular momentum is L=mvr only for special cases. Write it for the general case and you will get a more general result.
  4. Nov 25, 2013 #3
    Hmm, well this is why I thought it was for general U, maybe you can help me see what I did wrong:


    In polar, [itex]v^2=\dot{r}^2+r^2\dot{\theta}^2[/itex]


    And then in my text I found these equations:

    [itex]\frac{\partial L}{\dot{q}_i}=p_i[/itex]

    [itex]\frac{\partial L}{q_i}=\dot{p}_i[/itex]

    (Not quite sure why they're true yet)

    which implies:
    [itex]\frac{d}{dt}(\frac{\partial L}{\dot{q}_i})=\frac{\partial L}{q_i}[/itex]

    Applied to Lagrangian above:
    [itex]\frac{d}{dt}(\frac{\partial L}{\dot{\theta}})=\frac{\partial L}{\theta}[/itex]

    I do see the problem now though! In this expression, this is only velocity in the theta direction. It doesn't account for velocity in the r direction, so I can't use it in the OP. Thanks :)
    Last edited: Nov 25, 2013
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