Potential central field

  • Thread starter neworder1
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  • #1
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Homework Statement



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

[tex]V(r)=-\frac{\alpha}{r}+\frac{\beta}{r^{2}}[/tex]

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

Homework Equations





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?
 

Answers and Replies

  • #2
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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.
 
  • #3
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Could you be more specific? I don't see how we can get beyond what I've written above using the second order equation.
 
  • #4
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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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