Eur. Phys. J. Special Topics 229, 1405–1417 (2020)
https://doi.org/10.1140/epjst/e2020-900198-x

Regular Article

Some simple results about the Lambert problem

¹ IMCCE, CNRS-UMR8028 Observatoire de Paris, 77, Avenue Denfert-Rochereau, 75014 Paris, France
² Departamento de Matematica Aplicada, Facultad de Ciencias, Campus Universitario de Fuentenueva, Universidad de Granada, 18071 Granada, Spain

^a e-mail: Alain.Albouy@obspm.fr
^b e-mail: ajurena@ugr.es

Received: 16 September 2019
Received in final form: 22 January 2020
Published online: 29 May 2020

Abstract

We give simple proofs of some simple statements concerning the Lambert problem. We first restate and reprove the known existence and uniqueness results for the Keplerian arc. We also prove in some cases that the elapsed time is a convex function of natural parameters. Our statements and proofs do not distinguish between the three types of Keplerian conic section, elliptic, parabolic and hyperbolic. We also prove non-uniqueness results and non-convexity results. We do not develop any algorithm of resolution, limiting ourselves to such obviously useful a priori questions: How many solutions should we expect? Can we be sure that the Newton method will converge?