Study of relative equilibria in the planar n-body and n-vortex problem

• Grant number: 24/00971-0
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil
• Start date: September 01, 2024
• End date: November 30, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Principal Investigator: Clodoaldo Grotta Ragazzo
• Grantee: Clodoaldo Grotta Ragazzo
• Visiting researcher: Antonio Carlos Fernandes
• Visiting researcher institution: Instituto De Matemática E Computação/Imc/Unifei, Brazil
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

This project will be related to the current project "CATEDRAS FRANCO-BRASILEIRAS NO ESTADO DE SÃO PAULO" under the coordination of Clodoaldo Grotta Ragazzo and partnership with French host Alain Albouy, who has production related to central configurations, including with the visitor proponent , Antonio Carlos Fernandes. In addition to this, the FAPEMIG/Universal 2018 project, APQ 03149-18 under the coordination of the visiting researcher, Antonio Carlos Fernandes, which provides Chile/Brazil air tickets for the researcher Claudio Vidal from the research group in Sistemas Dinámicos y Aplicaciones at the Universidad del Bío-Bío de Concepción, Chile. In this sense, the insertion of a new collaboration for the Differential Equations and Applications research group at IME-USP, since professor Claudio Vidal will come to Brazil in August 2024 to collaborate with Antonio Carlos Fernandes and will pass through IME-USP.Antonio Carlos Fernandes is an associate professor at "Universidade Federal de Itajubá", "Minas Gerais". He holds degree in Physics UNIFEI (2008), a master's degree in Applied Mathematics IME-USP, advisor Clodoaldo Ragazzo (2009), a PhD in Applied Mathematics IME-USP, advisor Luis Fernando Mello (2011), post doctorate in "Universidad del Bío-Bío, Concepción, Chile" with Claudio Vidal (2018). He has been working on Celestial Mechanics and Central Configurations since 2007. He supervised seven Master's students, three completion works in mathematics and ten scientific initiation works in mathematics. He is currently a CNPq PQ2 productivity fellow.In this project we propose to study some aspects of the central configurations in the n-body problem and also in the n-plane vortex problem. More specifically, we intend to study: the characterization of the main properties of the central (n,2) and (n,3) stacked configurations in the n-body problem and the n-vortex problem; the uniqueness of the convex central configuration of four bodies, it is worth mentioning that this is a question that has been open since 1932 with the work of MacMillan and Bartky, published in Transactions of the American Mathematical Society, the uniqueness for quadrilaterals with diagonal symmetry was proved in 2008 by Albouy, Fu and Sun. The visiting proponent Antonio Carlos Fernandes together with Jaume Llibre and Luis Fernando Mello proved the conjecture in the case of symmetry, by reflection in a perpendicular bisector, that is, they proved the uniqueness of the isosceles trapezoid configuration as being the only central convex configuration that admits two pairs of adjacent equal masses; the characterization of central configurations of n bodies in which it is possible to vary the values of k masses, leaving all positions and the value of (n-k) masses fixed, this issue is linked to a type of rigidity and vertical degeneracies of the central configurations. It is worth telling that the issue of vertical degeneracies is problem 17 on the list of Cabral, Santos and Albouy, this problem now has a negative answer with the work "A limit of nonplanar 5-body central configurations is nonplanar" by Antonio Carlos Fernandes and Alain Albouy; the problem of non-collinear central super configurations in the case of 5 bodies, proposed by Alain Chenciner in 2003;As specific objectives, we point:Study planar and spatial central configurations;Study the case of planar central configurations formed by rings in the n-vortex problem;Study the uniqueness of convex central configurations of four bodies;Study co-circular central configurations;Describe the main properties of the (n,2) stacked and (n,3) stacked configurations;Study rigidity and vertical degeneracies of central configurations;Study cases of central super configuration with five bodies. (AU)