The Integral Manifolds of the 4 Body Problem with Equal Masses: Bifurcations at Relative Equilibria

In the N-body problem, it is classical that there are conserved quantities of center of mass, linear momentum, angular momentum and energy. The level sets \(\mathfrak {M}(c,h)\) of these conserved quantities are parameterized by the angular momentum c and the energy h, and are known as the integral manifolds. A long-standing goal has been to identify the bifurcation values, especially the bifurcation values of energy for fixed non-zero angular momentum, and to describe the integral manifolds at the regular values. Alain Albouy identified two categories of singular values of energy: those corresponding to bifurcations at relative equilibria; and those corresponding to “bifurcations at infinity”, and demonstrated that these are the only possible bifurcation values. This work completes the identification of bifurcations for the four-body problem with equal masses, confirming that, in this setting, Albouy’s necessary conditions for bifurcation are also sufficient conditions: bifurcations of the integral manifolds occur at all of the singular values of energy. A recent study examined the bifurcations at infinity; this work evaluates the four bifurcations at relative equilibria. To establish that the topology of the integral manifolds changes at each of these values, and to describe the manifolds at the regular values of energy, the homology groups of the integral manifolds are computed for the five energy regions on either side of the singular values. The homology group calculations establish that all four energy levels are indeed bifurcation values, and allows some of the global properties of the integral manifolds to be explored.