Abstract

A central configuration is a special arrangement of N point masses such that the gravitational acceleration vector of each mass due to he other masses is proportional to its position vector relative to the center of mass. The first examples of central configurations ate to the 18th century investigations of Euler and Lagrange. Since then, they have played an important role in the qualitative study of he dynamics. In addition to providing simple periodic solutions, they are relevant for studies of collision, scattering to infinity nd bifurcation of the integral manifolds

This course will cover some of these applications and then focus on methods for studying central configurations for their own sake. We will focus on questions of existence and enumeration of various types of central configurations, including algebraic-geometrical approaches to Smale’s 6th problem: Is the number of central configurations always finite?

Objective

To describe the main mechanisms leading to a fairly global description of the dynamics in conservative systems.

Topics

• Basic symplectic 2D maps. Return maps, standard, separatrix and Hénon maps. Invariant curves, invariant manifolds, splitting of separatrices, invariant Cantor sets, creation of chaos, Lyapunov exponents. Measure problems. Basic theoretical results and computational methods.
• Some key theoretical results: averaging theory under periodic and quasi-periodic excitation; normal forms, convergence problems and Gevrey properties; KAM theory; Nekhorosev theory, stable/unstable/centre manifolds.
• Symbolic and numeric tools for the computation of periodic and quasi-periodic solutions and invariant manifolds.
• Applications: Global behavior of orbits near the libration points in the RTBP; some problems and subproblems of the general 3-body system. Escape, capture and diffusion.

Abstract

The goal of these lectures is to study the periodic solutions of autonomous differential systems in Rⁿ via the averaging theory. This theory is based essentially in two theorems, we shall present them at any order and for arbitrary dimension. We shall apply this theory to the study of the periodic solutions of the van der Pol differential equation, Liénard differential systems, the Rossler differential system, and to some Hamiltonian systems.