Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 10C (Dynamics and Relativity)

A central force conserves angular momentumPut $u=1/r$. Since $\dot{\theta }=\ell u^2/m$,where primes now denote differentiation with respect to $\theta$. Substitution into the radial equation gives the Binet equation

For $V=-km/r$, one has $dV/dr=km{u}^2$, soAfter choosing the angular origin, the solution isFor $0\le e<1$, using $r=\sqrt{x^2+y^2}$ and $r+ex=p$ givesThis is an ellipse, with the force centre at one focus, as described by a Kepler orbit.

ForBinet's equation becomesA circular orbit has constant $u=u_c$ and therefore satisfiesThe right side has maximum $4/(27r_0)$ at $u_c=1/(3r_0)$. Thus there are two circular orbits when $0<K<4/(27r_0)$, one marginal orbit at equality, and none above it.

Writing $u=u_c+\eta$ and linearizing givesHence the outer orbit $u_c<1/(3r_0)$, equivalently $r_c>3r_0$, is stable; the inner orbit is unstable. These are the circular-orbit branches of the shifted inverse-radius potential.

To first order in $r_0$,The expanded orbit equation is thereforeIts required family of solutions iswith any $0<A<1$. Successive periapses occur when $\omega \theta$ changes by $2\pi$, so their angular separation isThe periapsis advance per revolution is consequently $2\pi K{r}_0+O(r_0^2)$.