PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Gravitation and central-force motion · Gravitation and central-force motion
Action and Symmetry, Ch The Kepler problem redone: reduction, quadrature and the Runge-Lenz vector
Problem
An attractive central force has magnitude F(r)=Cr13 (C>0), so β=13.
((a)) Show that the apsidal angle of a nearly circular orbit in this force is a rational multiple of π, and give its value.
((b)) A colleague concludes from part (a) that, since the near-circular apsidal angle is a rational multiple of π, every bound orbit in this force must close. Predict whether this conclusion is correct, and say what Bertrand's theorem actually implies for this β.