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Potential central field

  1. Nov 18, 2007 #1
    1. The problem statement, all variables and given/known data

    The problem is to find the motion of a body in a central potential field with potential given by:


    where [tex]\alpha[/tex] and [tex]\beta[/tex] are positive constants.

    2. Relevant equations

    3. The attempt at a solution

    I used the fact that energy and angular momentum are conserved in this field, and after separating variables in the equation for [tex]\dot{\vec{r}}[/tex] I got an integral of the form: ([tex]\phi[/tex] is the angle)

    [tex]\phi = \int{\frac{dr}{\sqrt{Ar^{3}-Br^{2}+C}}}[/tex]

    where A, B, C are constants dependent on mass, energy and angular momentum of the body.

    Is there a simpler method to find the motion [tex]r(\phi)[/tex], without having to calculate such awful integrals? And if not, how to calculate it?
  2. jcsd
  3. Nov 18, 2007 #2
    Yes, I think there is. Note that you have two differential equations: one first order and one second order (the Lagrange equation). Hint: use the second order. But since you are interested in the shape, you need to change from time derivatives to derivatives wrt [itex]\phi[/itex]. Question: what is the relationship between [itex]\dot{r}[/itex] and [itex]r'(\phi)[/itex]? Answering this question will lead you to a differential equation for your trajectory.
  4. Nov 18, 2007 #3
    Could you be more specific? I don't see how we can get beyond what I've written above using the second order equation.
  5. Nov 18, 2007 #4
    You need to use the fact that [itex]\dot{r} = \dot{\phi}r'(\phi)[/itex]. Use this to eliminate all derivatives wrt time in your Lagrange equation. But before you do, what is [itex]\dot{\phi}[/itex]?
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