EZEQUIEL MADERNA
Universidad de la República, Uruguay
Abstract :
The extension of the Aubry-Mather theory for convex Lagrangians to non-compact man ifolds, and then further allowing the Lagrangian to have singularities, naturally led to the notion of critical energy level in these systems, and in particular to the verification of the existence of weak global solutions of the critical Hamilton-Jacobi equation. These are the socalled weak KAM solutions. From this approach applied to the classical N-body problem [Ma], the abundance of completely parabolic motions emerged first. The critical level is the hinge between, on the one hand, the dynamics with chances of recurrence, and on the other, those that present expansion or scattering.
More recently, in work with A. Venturelli [MaVe], we have distilled the basis for the application of the Hamilton-Jacobi method in almost all Newtonian-type gravitational problems, using the now well known notion of viscosity solution. It is obtained, for example, that the Busemann functions associated with a given limit shape are of this type of solutions, and from them the existence of completely hyperbolic motions is deduced for positive energy levels, with arbitrary initial positions and arbitrary limit shapes.
In this course I will develop this approach from a geometric point of view. More precisely, the keys to this technique will be seen in terms of geodesic rays of the classic Jacobi Maupertuis metric, on which we are actively working today (see [PeSa][BuMa]).
[Ma] E. Maderna. On weak KAM theory for N–body problems, Ergod. Th. & Dynam. Sys. 32 (2012), 1019–1041.
[MaVe] E. Maderna, A. Venturelli. Viscosity solutions and hyperbolic motions: a new PDE method for the N–body problem, Annals of Mathematics 192 n.2 (2020), 499–550. [BuMa] J. M. Burgos, E. Maderna. On the geodesic rays of the positive energy levels in the Newtonian N–body problem, in preparation.
[PeSa] B. Percino, H. Sánchez-Morgado. Busemann functions for the N–body problem, Arch. Rational Mech. Anal. 213 (2014), 981–991.