From Babylonian lunar observations to Floquet multipliers and Conley–Zehnder indices.

The article focuses on the analysis of Hill's lunar problem, a limiting case of the restricted three-body problem modeling the Sun–Earth–Moon system, specifically examining a planar direct periodic orbit discovered by George William Hill in 1878. It establishes a connection between the classical lunar periods—the anomalistic and draconitic months—and the planar and spatial Floquet multipliers and Conley–Zehnder indices associated with Hill's orbit, providing a geometric interpretation of these astronomical phenomena. Using symplectic geometry and bifurcation theory, the study analytically derives the Conley–Zehnder indices for fundamental families of planar direct and retrograde periodic orbits (families g and f) emerging from the rotating Kepler problem at low energies, and extends these results to spatial orbits including collision orbits. The work also relates these indices to the rotation numbers that determine the anomalistic and draconitic periods, yielding numerical approximations closely matching historical Babylonian and modern lunar period values. Finally, the article discusses the regularization of the problem, the Morse–Bott structure of the action functional, and the bifurcation scenario of periodic orbits, highlighting open problems in analytically establishing Hill’s variational orbit.