The Conley–Zehnder indices of the spatial Hill three-body problem

Abstract

We explore the interaction between the Conley–Zehnder index and bifurcation points of symmetric planar as well as spatial periodic orbits in the spatial Hill three-body problem. We start with the fundamental families of planar periodic orbits which are those of direct (family g) and retrograde periodic orbits (family f). Since the spatial system is invariant under a symplectic involution, whose fixed point set corresponds to the planar problem, planar orbits have planar and spatial Floquet multipliers, and planar and spatial Conley–Zehnder indices. When the Floquet multipliers move through a root of unity, new families of periodic orbits bifurcate and the index jumps. For very low energies, the families g and f arise dynamically from the rotating Kepler problem, and in a recent work (Aydin From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices) we determined analytically their indices. By their numerical continuations for higher energies, we determine the index of various families of planar and spatial periodic orbits bifurcating from g and f. Since these families can bifurcate again and meet each other, this procedure can get complicated. This index leads to a grading on local Floer homology. Since the local Floer homology and its Euler characteristic stay invariant under bifurcation, the index provides important information about the interconnectedness of such families, which we illustrate in form of bifurcation graphs. Since the solutions of Hill’s system may serve as orbits for space mission design or astronomical observations, our results promote the interaction between Symplectic Geometry and practical problems.