EDDALY GUERRA VELASCO 

Universidad Autónoma de Chiapas, México 

Abstract :

For many reasons, central configurations play a fundamental role in the study of the dynamics of the N-body problem. First, they are the only configurations through which some orbits pass that we can explicitly describe. Second, they arise naturally as limit shapes of total collision or completely parabolic expansion motions. They can be characterized as critical points of the Newtonian potential, and in this sense their determination gives rise to complex systems of polynomial equations, which appear treated from the beginning by Euler (for the case of three masses in the line [Eu]), until in the recent resolution in 2012 by Albouy and Kaloshin [AK] of the finitude problem (Smale problem for the 21st century [Sm]) for the case of 5 bodies in the plane. They also can be defined for all systems governed by homogeneous interaction potentials. 

The main objective of this course is to introduce the student, through the construction of homographic orbits (and fitting in a certain way the Keplerian orbits within a system with three or more bodies), to the handling of the equations of motion, understanding the complexity of the dynamics, which is fundamental for the good assimilation of several of the advanced courses offered in this school. On the other hand, this introductory and preparatory course allows by itself to invite the student to a vast topic of current research. 

The main reference for this course will be the well known notes by Rick Moeckel [Mo]. 

[Eu] L. Euler, De motu rectilineo trium corporum se mutuo attrahentium. Novi commen tarii academiae scientiarum Petropolitanae 11, 144–151 (1765). (read at St Petersburg in december 1763. Also in Opera Omnia S. 2, vol. 25, 281–289) 

[AK] A. Albouy and V. Kaloshin, Finiteness of central configurations of five bodies in the plane. Ann. Math. (2) 176, 535–588 (2012) 

[Sm] S. Smale, Mathematical problems for the next century. Math. Intell. 20 (1998),