On the minimality of Keplerian arcs with fixed negative energy

TL;DR

This paper revisits classical results on the local minimality of Keplerian orbits with fixed negative energy, using Morse theory and conjugate point analysis to deepen understanding of their variational properties.

Contribution

It provides a new proof of the minimality of Keplerian arcs employing Morse index theory and conjugate point characterization, enhancing classical results.

Findings

Keplerian arcs with fixed negative energy are locally minimal between two points.

Morse index theorem is effective in analyzing the minimality of Keplerian solutions.

Conjugate points correspond to geodesic bifurcations in the energy functional.

Abstract

We revisit a classical result by Jacobi on the local minimality, as critical points of the corresponding energy functional, of fixed-energy solutions of the Kepler equation joining two distinct points with the same distance from the origin. Our proof relies on the Morse index theorem, together with a characterization of the conjugate points as points of geodesic bifurcation.