Dynamics | Part IA, 2001

Derive the equation

d2udθ2+u=f(u)mh2u2\frac{d^{2} u}{d \theta^{2}}+u=\frac{f(u)}{m h^{2} u^{2}}

for the motion of a particle of mass mm under an attractive central force ff, where u=1/ru=1 / r and rr is the distance of the particle from the centre of force, and where mhm h is the angular momentum of the particle about the centre of force.

[Hint: you may assume the expressions for the radial and transverse accelerations in the form r¨rθ˙2,2r˙θ˙+rθ¨\ddot{r}-r \dot{\theta}^{2}, 2 \dot{r} \dot{\theta}+r \ddot{\theta}.]

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