The integral manifolds of the 4 body problem with equal masses: bifurcations at infinity (Q6612545)

This work furthers the investigation of the integral manifolds in the spatial \( N \)-body problem. In this classical setting, the conserved quantities include the center of mass, linear momentum, angular momentum, and energy. The integral manifolds correspond to the level sets \({\mathfrak M}(c, h)\) of these conserved quantities, which are parameterized by the angular momentum \(c\) and the energy \(h\). In the spatial problem, they form a family of (\(6N -10\))-dimensional manifold.\N\NThe paper under review examines the bifurcations of the manifolds \({\mathfrak M}(c, h)\) for a fixed non-zero angular momentum and provides a detailed description of the integral manifolds at the regular values in the four-body problem with equal masses. The bifurcation analysis is carried out using algebraic topology, with a particular emphasis on homology theory. Starting from the algebraic equations defining the integral manifolds, the homology groups of the associated spaces are computed. Variations in these groups are then used to detect changes in the topology of the manifolds.\N\NThis work takes directly on prior contributions to the field, notably those of \textit{A. Albouy} [Invent. Math. 114, No. 3, 463--488 (1993; Zbl 0801.70008)] and the author's earlier work [J. Dyn. Differ. Equations 35, No. 1, 1--68 (2023; Zbl 1525.70022)]. The foundational framework established in these works is concisely reviewed in Section 2. In particular, Albouy classified singular energy values into two categories: bifurcations at relative equilibria and ``bifurcations at infinity'', demonstrating that these are the only possible types of bifurcation values. In this context, the author confirms the existence of four singular values corresponding to bifurcations at infinity. To establish that the topology of the integral manifolds changes at each of these values and to describe the manifolds at the regular energy values, the homology groups of the integral manifolds are computed for the five energy regions delineated by the singular values. Those changes in the homology groups confirm that the set of allowable positions undergoes changes at the bifurcations at relative equilibria. This points up the different nature of the bifurcations at infinity, in comparison to the bifurcations at relative equilibria.