Paper 4, Section II, B

(a) A particle $P$ of unit mass moves in a plane with polar coordinates $(r, \theta)$. You may assume that the radial and angular components of the acceleration are given by $\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right)$, where the dot denotes $d / d t$. The particle experiences a central force corresponding to a potential $V=V(r)$.

(i) Prove that $l=r^{2} \dot{\theta}$ is constant in time and show that the time dependence of the radial coordinate $r(t)$ is equivalent to the motion of a particle in one dimension $x$ in a potential $V_{\text {eff }}$ given by

$V_{\text {eff }}=V(x)+\frac{l^{2}}{2 x^{2}}$

(ii) Now suppose that $V(r)=-e^{-r}$. Show that if $l^{2}<27 / e^{3}$ then two circular orbits are possible with radii $r_{1}<3$ and $r_{2}>3$. Determine whether each orbit is stable or unstable.

(b) Kepler's first and second laws for planetary motion are the following statements:

K1: the planet moves on an ellipse with a focus at the Sun;

K2: the line between the planet and the Sun sweeps out equal areas in equal times.

Show that K2 implies that the force acting on the planet is a central force.

Show that K2 together with $\mathbf{K 1}$ implies that the force is given by the inverse square law.

[You may assume that an ellipse with a focus at the origin has polar equation $\frac{A}{r}=1+\varepsilon \cos \theta$ with $A>0$ and $0<\varepsilon<1$.]

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