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In this thread, I propose a very naive, homemade model of particle behavior in an atmosphere.
Assume that the Earth and its atmosphere rotate as a unified rigid body about a fixed axis with constant angular velocity ##\boldsymbol\omega##. Assume that a particle of mass ##m## moves in a plane perpendicular to the vector ##\boldsymbol\omega## and containing the center of the Earth.
The particle is subjected to a gravitational force and a drag force
$$\boldsymbol F=-\gamma \boldsymbol v_{rel},\quad \boldsymbol v_{rel}=\boldsymbol v-\boldsymbol \omega\times\boldsymbol r,\quad \gamma>0.$$
Here, ##\boldsymbol v## is the particle's velocity, and ##\boldsymbol v_{rel}## is its velocity relative to the atmosphere.
##\boldsymbol r## stands for the position vector of the particle relative to the Earth's center (the origin).
Introduce fixed polar (inertial) coordinates ##(r,\varphi)## such that ##\boldsymbol r=r\boldsymbol e_r##
and
$$\boldsymbol v=\dot r\boldsymbol e_r+r\dot\varphi\boldsymbol e_\varphi.$$
Thus, in these fixed polar coordinates, the Lagrangian of the system (excluding the non-conservative drag force) is
$$L=\frac{m}{2}\left(\dot r^2+r^2\dot\varphi^2\right)+\frac{G}{r},\quad G>0.$$
The Lagrange equations of the second kind are written as
$$\frac{d}{dt}\frac{\partial L}{\partial \dot r}-\frac{\partial L}{\partial r}=Q_r,\quad
\frac{d}{dt}\frac{\partial L}{\partial \dot \varphi}-\frac{\partial L}{\partial \varphi}=Q_\varphi,$$
where ##Q_r## and ##Q_\varphi## are the non-conservative generalized forces defined by
$$Q_r=\left(\frac{\partial\boldsymbol v}{\partial \dot r},\boldsymbol F\right)=-\gamma\dot r,\quad
Q_\varphi=\left(\frac{\partial\boldsymbol v}{\partial \dot \varphi},\boldsymbol F\right)=-\gamma r^2(\dot\varphi-\omega).$$
I think it would be interesting if someone could plot a few trajectories numerically and share the images here
Assume that the Earth and its atmosphere rotate as a unified rigid body about a fixed axis with constant angular velocity ##\boldsymbol\omega##. Assume that a particle of mass ##m## moves in a plane perpendicular to the vector ##\boldsymbol\omega## and containing the center of the Earth.
The particle is subjected to a gravitational force and a drag force
$$\boldsymbol F=-\gamma \boldsymbol v_{rel},\quad \boldsymbol v_{rel}=\boldsymbol v-\boldsymbol \omega\times\boldsymbol r,\quad \gamma>0.$$
Here, ##\boldsymbol v## is the particle's velocity, and ##\boldsymbol v_{rel}## is its velocity relative to the atmosphere.
##\boldsymbol r## stands for the position vector of the particle relative to the Earth's center (the origin).
Introduce fixed polar (inertial) coordinates ##(r,\varphi)## such that ##\boldsymbol r=r\boldsymbol e_r##
and
$$\boldsymbol v=\dot r\boldsymbol e_r+r\dot\varphi\boldsymbol e_\varphi.$$
Thus, in these fixed polar coordinates, the Lagrangian of the system (excluding the non-conservative drag force) is
$$L=\frac{m}{2}\left(\dot r^2+r^2\dot\varphi^2\right)+\frac{G}{r},\quad G>0.$$
The Lagrange equations of the second kind are written as
$$\frac{d}{dt}\frac{\partial L}{\partial \dot r}-\frac{\partial L}{\partial r}=Q_r,\quad
\frac{d}{dt}\frac{\partial L}{\partial \dot \varphi}-\frac{\partial L}{\partial \varphi}=Q_\varphi,$$
where ##Q_r## and ##Q_\varphi## are the non-conservative generalized forces defined by
$$Q_r=\left(\frac{\partial\boldsymbol v}{\partial \dot r},\boldsymbol F\right)=-\gamma\dot r,\quad
Q_\varphi=\left(\frac{\partial\boldsymbol v}{\partial \dot \varphi},\boldsymbol F\right)=-\gamma r^2(\dot\varphi-\omega).$$
I think it would be interesting if someone could plot a few trajectories numerically and share the images here
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