The discussion revolves around the stability of orbits under a central force described by the equation ##\mathbf{F}(\mathbf{r}) = -\frac{\mu}{r^2} e^{-kr} \hat{r}##. The original poster presents a scenario where a particle moves in a circular orbit and explores the effects of small perturbations on this motion.
The discussion is ongoing, with participants seeking clarification on mathematical steps and the implications of their findings. Some participants express uncertainty about the correctness of certain derivations and the physical interpretation of the results, while others suggest potential corrections or alternative perspectives.
Participants are working under the constraints of a homework assignment, which may limit the information available for discussion. There are indications of confusion regarding the linearization of exponential terms and the identification of equilibrium points in the perturbed system.
Thanks voko,voko said:I just say that this shorter or longer duration is my time unit. They are completely arbitrary to begin with. But they do correspond to some physical process. A useful illustration is the year. This is basically a unit such that the orbital period of the Earth is 1. The month is the (roughly) orbital period of the Moon. The day is a unit such that the period of the Earth rotation is 1. We use all these units daily (oops, just did again), yet they were specifically chosen so that some particular celestial motions had periods = 1.
Note I am talking about periods here. In my previous post I was talking about the angular speed. There is no real difference, it all depends on what we consider more basic, the frequency or the period.