这是一份bath巴斯大学PH30111作业代写的成功案

$$
\begin{aligned}
&r=R+x \
&\theta=\Omega t+y
\end{aligned}
$$
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
$$
\kappa^{2}=\frac{\partial^{2} U(R, 0)}{\partial r^{2}}+3 \Omega^{2}=R \frac{\mathrm{d} \Omega^{2}}{\mathrm{~d} R}+4 \Omega^{2} .
$$