# 银河系和宇宙学入门|PH30111 Galaxies and introduction to cosmology代写

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\begin{aligned} &r=R+x \ &\theta=\Omega t+y \end{aligned}
where $\Omega$ is the angular velocity for a circular path:
$$\Omega^{2}=\frac{1}{R} \frac{\partial U(R, 0)}{\partial r}$$
In polar coordinates the equations of motion are written as\begin{aligned} \ddot{r}-\dot{\theta}^{2} r &=-\frac{\partial U}{\partial r}, \ r \ddot{\theta}+2 \dot{r} \dot{\theta} &=0 \ \ddot{z} &=-\frac{\partial U}{\partial z} . \end{aligned}

## PH30111 COURSE NOTES ：

and inserting this into the first equation, we have
$$\ddot{x}-2 \Omega(a-2 x \Omega)-\Omega^{2} x=-x \frac{\partial^{2} U(R, 0)}{\partial r^{2}} .$$
This equation can be written in the form
$$\ddot{x}+\kappa^{2}\left(x-x_{0}\right)=0$$
and
$$\dot{y}=-\frac{2 \Omega x}{R},$$
with
$$\kappa^{2}=\frac{\partial^{2} U(R, 0)}{\partial r^{2}}+3 \Omega^{2}=R \frac{\mathrm{d} \Omega^{2}}{\mathrm{~d} R}+4 \Omega^{2} .$$