Central Force Integral: Finding the Solution | Starting Guide

[cscott]

Homework Statement

$$\int dr \left[\alpha + \frac{\beta}{r^2}\right]^{-1/2}$$

How can I get started on this? Thanks.

 

[dirk_mec1]

Multiply numerator and denominator with r and use substitution rule.

 

[cscott]

so if $\alpha = 2E/\mu$ and $\beta = L^2\alpha^2/\mu^2$ (not the same alpha, sorry) and bounds [r0,r] I should get:

$$\frac{\mu}{2E} \left[\left(\frac{2E}{\mu}r^2 + \frac{L^2\alpha^2 }{\mu^2}\right)^{1/2} - \left(\frac{2E}{\mu}r_0^2 + \frac{L^2\alpha^2 }{\mu^2}\right)^{1/2}\right]$$

and this is equal to time so solving r(t) gives a quadratic. Does this make sense for central force motion where $r = ke^{-\alpha\theta}$?