Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Orbital stability and fictitious potential energy with a change of convention

  1. Mar 27, 2016 #1
    The condition for a stable orbit is given by (3.42), where ##V'## is the fictitious potential energy (potential energy of the corresponding fictitious one-dimensional problem) and ##r_0## is the radius of the circular orbit. The result ##n>-3## is obtained by using the convention that positive forces point radially outwards. This result should be independent of the convention used. So if we instead take positive forces to point radially inwards, we should get the same result. However, I get a different result, i.e., ##n<-3##.

    Under the new convention, we have ##f=kr^n##. After substituting into (3.43), we have ##knr^{n-1}<-3kr^{n-1}## or ##n<-3## (keeping in mind that ##f(r_0)=\frac{l^2}{mr_0^3}## under the new convention).

    What's wrong?

    Screen Shot 2016-03-28 at 1.17.35 am.png
    Screen Shot 2016-03-28 at 2.55.53 am.png
    Screen Shot 2016-03-28 at 1.17.56 am.png

    EDIT: I found the mistake: (3.42) and (3.43) need to be modified when the convention is changed.

    Under the new convention, ##f=\frac{\partial V}{\partial r}## instead. So (3.12) becomes ##m\ddot{r}-\frac{l^2}{mr^3}=-f(r)##. And the fictitious force ##-f'## becomes ##-f'=-f+\frac{l^2}{mr^3}##. (3.42) becomes ##\frac{\partial^2 V'}{\partial r^2}|_{r=r_0}=\frac{\partial f}{\partial r}|_{r=r_0}+\frac{3l^2}{mr_0^4}>0##.

    Screen Shot 2016-03-28 at 1.13.47 am.png
    Screen Shot 2016-03-28 at 1.14.04 am.png
    Screen Shot 2016-03-28 at 2.40.40 am.png
    Screen Shot 2016-03-28 at 2.40.58 am.png
     
    Last edited: Mar 27, 2016
  2. jcsd
  3. Apr 1, 2016 #2
    Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Orbital stability and fictitious potential energy with a change of convention
  1. Stability of Orbits (Replies: 1)

Loading...