Orateur
Description
(with Holger Dullin).
In any motion of the n-body problem, seen in the Galilean frame of the center of mass, the energy H and the angular momentum vector L are constant.
The integral manifolds are the submanifolds of the phase space defined by fixing H and L. If we fix L and let H vary, the submanifold will change topology at values which we call singular. The critical values are singular values, but for example H=0 is a singular value which is not critical, already in the 2-body problem.
In [1], developing works by Smale, Cabral, Easton and Simó, I gave a list of values which includes all the singular values in the 3-dimensional n-body problem. I conjectured that this list does not contain artefacts but only singular values. This was later proved in the 3-body case by McCord, Meyer and Wang, by homology computations.
What is still missing is a general argument in the n-body case proving that the values in the list are all significant.
In an ongoing work with Holger Dullin we convinced ourselves that some specific results about the gradient flow of H restricted to L=L_0 and some general results about the critical points at infinity should give such a general argument. I will explain examples of such results.
[1] Alain Albouy, Integral manifolds of the N-body problem
Inventiones Mathematicae, 114 (1993), 463-488
[2] Andreas Knauf and Nikolay Martynchuk, Topology change of level sets in Morse theory, Arkiv for Matematik, 58 (2020), 333–356
[3] Christopher K. McCord, The Integral Manifolds of the 4 Body Problem with Equal Masses: Bifurcations at Relative Equilibria, Journal of Dynamics and Differential Equations, (2024)
[4] Alain Albouy, Holger R. Dullin, Bounded orbits for three bodies in ℝ^4, Geometric Mechanics, 1 (2024)
