Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2010

A particle of unit mass moves in a plane with polar coordinates (r,θ)(r, \theta) and components of acceleration (r¨rθ˙2,rθ¨+2r˙θ˙)\left(\ddot{r}-r \dot{\theta}^{2}, r \ddot{\theta}+2 \dot{r} \dot{\theta}\right). The particle experiences a force corresponding to a potential Q/r-Q / r. Show that

E=12r˙2+U(r) and h=r2θ˙E=\frac{1}{2} \dot{r}^{2}+U(r) \quad \text { and } \quad h=r^{2} \dot{\theta}

are constants of the motion, where

U(r)=h22r2QrU(r)=\frac{h^{2}}{2 r^{2}}-\frac{Q}{r}

Sketch the graph of U(r)U(r) in the cases Q>0Q>0 and Q<0Q<0.

(a) Assuming Q>0Q>0 and h>0h>0, for what range of values of EE do bounded orbits exist? Find the minimum and maximum distances from the origin, rminr_{\min } and rmaxr_{\max }, on such an orbit and show that

rmin+rmax=QE.r_{\min }+r_{\max }=\frac{Q}{|E|} .

Prove that the minimum and maximum values of the particle's speed, vminv_{\min } and vmaxv_{\max }, obey

vmin+vmax=2Qhv_{\min }+v_{\max }=\frac{2 Q}{h}

(b) Now consider trajectories with E>0E>0 and QQ of either sign. Find the distance of closest approach, rminr_{\min }, in terms of the impact parameter, bb, and vv_{\infty}, the limiting value of the speed as rr \rightarrow \infty. Deduce that if bQ/v2b \ll|Q| / v_{\infty}^{2} then, to leading order,

rmin2Qv2 for Q<0,rminb2v22Q for Q>0r_{\min } \approx \frac{2|Q|}{v_{\infty}^{2}} \text { for } Q<0, \quad r_{\min } \approx \frac{b^{2} v_{\infty}^{2}}{2 Q} \text { for } Q>0

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