Paper 4, Section I, B

Dynamics and Relativity | Part IA, 2011

The motion of a planet in the gravitational field of a star of mass MM obeys

d2r dt2h2r3=GMr2,r2dθdt=h\frac{\mathrm{d}^{2} r}{\mathrm{~d} t^{2}}-\frac{h^{2}}{r^{3}}=-\frac{G M}{r^{2}}, \quad r^{2} \frac{\mathrm{d} \theta}{\mathrm{d} t}=h

where r(t)r(t) and θ(t)\theta(t) are polar coordinates in a plane and hh is a constant. Explain one of Kepler's Laws by giving a geometrical interpretation of hh.

Show that circular orbits are possible, and derive another of Kepler's Laws relating the radius aa and the period TT of such an orbit. Show that any circular orbit is stable under small perturbations that leave hh unchanged.

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