Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2013

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

(i) Prove that l=r2θ˙l=r^{2} \dot{\theta} is constant in time and show that the time dependence of the radial coordinate r(t)r(t) is equivalent to the motion of a particle in one dimension xx in a potential Veff V_{\text {eff }} given by

Veff =V(x)+l22x2V_{\text {eff }}=V(x)+\frac{l^{2}}{2 x^{2}}

(ii) Now suppose that V(r)=erV(r)=-e^{-r}. Show that if l2<27/e3l^{2}<27 / e^{3} then two circular orbits are possible with radii r1<3r_{1}<3 and r2>3r_{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 K1\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 Ar=1+εcosθ\frac{A}{r}=1+\varepsilon \cos \theta with A>0A>0 and 0<ε<10<\varepsilon<1.]

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