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

1. Mar 27, 2016

### Happiness

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?

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 $\frac{\partial^2 V'}{\partial r^2}|_{r=r_0}=\frac{\partial f}{\partial r}|_{r=r_0}+\frac{3l^2}{mr_0^4}>0$.

Last edited: Mar 27, 2016
2. Apr 1, 2016