Dynamics | Part IA, 2003

Let (r,θ)(r, \theta) be plane polar coordinates and er\mathbf{e}_{r} and eθ\mathbf{e}_{\theta} unit vectors in the direction of increasing rr and θ\theta respectively. Show that the velocity of a particle moving in the plane with polar coordinates (r(t),θ(t))(r(t), \theta(t)) is given by

x˙=r˙er+rθ˙eθ,\dot{\mathbf{x}}=\dot{r} \mathbf{e}_{r}+r \dot{\theta} \mathbf{e}_{\theta},

and that the unit normal n\mathbf{n} to the particle path is parallel to

rθ˙err˙eθr \dot{\theta} \mathbf{e}_{r}-\dot{r} \mathbf{e}_{\theta} \text {. }

Deduce that the perpendicular distance pp from the origin to the tangent of the curve r=r(θ)r=r(\theta) is given by

r2p2=1+1r2(drdθ)2\frac{r^{2}}{p^{2}}=1+\frac{1}{r^{2}}\left(\frac{d r}{d \theta}\right)^{2}

The particle, whose mass is mm, moves under the influence of a central force with potential V(r)V(r). Use the conservation of energy EE and angular momentum hh to obtain the equation

1p2=2m(EV(r))h2\frac{1}{p^{2}}=\frac{2 m(E-V(r))}{h^{2}}

Hence express θ\theta as a function of rr as the integral

θ=hr2dr2m(EVeff(r))\theta=\int \frac{h r^{-2} d r}{\sqrt{2 m\left(E-V_{\mathrm{eff}}(r)\right)}}


Veff(r)=V(r)+h22mr2V_{\mathrm{eff}}(r)=V(r)+\frac{h^{2}}{2 m r^{2}}

Evaluate the integral and describe the orbit when V(r)=cr2V(r)=\frac{c}{r^{2}}, with cc a positive constant.

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