The discussion centers on the analysis of a particle of mass m moving under a central force defined by the equation ##\mathbf{F}(\mathbf{r}) = -\frac{\mu}{r^2} e^{-kr} \hat{r}##. The participants confirm that the condition for circular motion is ##h^2 = (a\mu/m)e^{-ka}## and derive the orbit equation for ##u(\theta)## as $$u'' + u = \frac{1}{a}e^{-k(1/u - a)}$$. They explore perturbations in the orbit, leading to the equation $$\ddot{p} + \frac{\mu e^{-ka}}{ma^3} p = \frac{k\mu e^{-ka}}{ma^2}p$$, ultimately identifying the angular frequency as $$w = \sqrt{1-ka}$$, which relates to the period of the perturbed motion.
PREREQUISITESStudents and researchers in physics, particularly those focusing on classical mechanics, orbital dynamics, and mathematical modeling of physical systems.
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.