What is the method for finding motion in a central potential field?

[neworder1]

Homework Statement

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

$$V(r)=-\frac{\alpha}{r}+\frac{\beta}{r^{2}}$$

where $$\alpha$$ and $$\beta$$ 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 $$\dot{\vec{r}}$$ I got an integral of the form: ($$\phi$$ is the angle)

$$\phi = \int{\frac{dr}{\sqrt{Ar^{3}-Br^{2}+C}}}$$

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 $$r(\phi)$$, without having to calculate such awful integrals? And if not, how to calculate it?

 

[noospace]

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 $\phi$. Question: what is the relationship between $\dot{r}$ and $r'(\phi)$? Answering this question will lead you to a differential equation for your trajectory.

 

[neworder1]

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

 

[jdstokes]

You need to use the fact that $\dot{r} = \dot{\phi}r'(\phi)$. Use this to eliminate all derivatives wrt time in your Lagrange equation. But before you do, what is $\dot{\phi}$?