Minimizers for the Kepler problem (Q2301839)

The author characterizes the minimizing geodesics for the Kepler problem endowed with the Jacobi-Maupertuis metric, in all the cases. Jacobi stated without proof in 1836 a condition under which the Keplerian elliptic orbit is the minimizer. A proof was given in 1871 by I. Todhunter, who also discovered the `turnpike' minimizers, which are made of rectilinear and circular parts. The author gives a precise review as well as new and simple proofs, based on a Marchal type lemma in the positive energy case, and on Lambert's theorem in the negative energy case.