Lambert's Theorem: Geometry or Dynamics?

TL;DR

This paper presents two new proofs of Lambert's theorem on Keplerian arcs, one inspired by Hamilton's variational approach and another based on geometric affine transformations, also discussing related topological properties.

Contribution

It introduces two novel proofs of Lambert's theorem, enhancing understanding through variational and geometric methods, and explores related topological and affine transformation properties.

Findings

Two new proofs of Lambert's theorem are provided.

Keplerian arcs related by Lambert's theorem share the same topology.

Related results on conic sections and affine transformations are discussed.

Abstract

Lambert's theorem (1761) on the elapsed time along a Keplerian arc drew the attention of several prestigious mathematicians. In particular, they tried to give simple and transparent proofs of it (see our timeline \S 9). We give two new proofs. The first one (\S 4) goes along the lines of Hamilton's variational proof in his famous paper of 1834, but we shorten his computation in such a way that the hypothesis is now used without redundancy. The second (\S 6) is among the few which are close to Lambert's geometrical proof. It starts with the new remark that two Keplerian arcs related by the hypothesis of Lambert's theorem correspond to each other through an affine map. We also show (\S 7) that despite the singularities due to the occurrence of collisions, the classes of arcs related by Lambert's theorem all have the same topology. We give (\S 8) some simple related results about conic sections and affine transformations.