Geometrical Themes Inspired by the N-body Problem

These are the notes of a series of lectures on ordinary differential equations in the complex domain delivered at the “Seventh Minimeeting in Differential Geometry” at CIMAT, in Guanajuato, Mexico, in 2015. We use geometric structures on curves as a setting to present some historical results of the theory and as a tool for a better understanding of some classical equations.

Deleting collisions from the configuration space of the planar N-body problem yields a space with a large interesting set of free homotopy classes of loops, classes which are encoded by “syzygy sequences” when N = 3. This expository piece centers on the question “ Is every free homotopy class of loops realized by a periodic solution to the problem?” We report on the recent affirmative answer (Moeckel and Montgomery, Nonlinearity 28:1919–1935, 2015) for the case of non-zero but small angular momentum and three equal or near-equal masses. The key tool is the McGehee blow-up (McGehee, Invent Math 27:191–227, 1974) as implemented by Rick Moeckel in the 1980s. After recounting some history and motivation, about a third of this article exposes the blow-up method. We use an energy balance under scaling transformations to motivate McGehee’s blow-up transformation. We give an explicit description of the blown-up and reduced phase space for the planar N-body problem, N ≥ 3 as a complex vector bundle over \([0, \infty ) \times \mathbb {C} \mathbb {P} ^{N-2}\). We end by returning to the angular momentum zero case where we conjecture the answer is ‘no’. We support this conjecture by recent work of Jackman and Rose (The Binary Returns!. arxiv:1512.01852, 2015; Geometric phase and periodic orbits of the equal-mass, planar three-body problem with vanishing angular momentum, Ph.D. Thesis, School of Mathematics and Statistics University of Sydney, 2015).

In this note we present a brief introduction to Lagrangian Floer homology and its relation to the solution to the Arnol’d Conjecture, on the minimal number of non-degenerate fixed points of a Hamiltonian diffeomorphism. We start with the basic definition of a critical point on smooth manifolds, in order to sketch some aspects of Morse theory. Introduction to the basics concepts of symplectic geometry are also included with the idea of understanding the statement of the Arnol’d Conjecture and how it is related to the intersection of Lagrangian submanifolds.