Instability of equilibrium points in Lagrangian systems

• Grant number: 10/20059-1
• Support Opportunities: Regular Research Grants
• Start date: April 01, 2011
• End date: March 31, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Ricardo dos Santos Freire Júnior
• Grantee: Ricardo dos Santos Freire Júnior
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

In this project we search for sufficient conditions fpr instability of equilibrium points in particular lagrangian systems where the potencial energy does not have a minimum at the equilibrium point and such fact can be detected through the study of its taylor polynomial of order $k$ or, more precisely, its jet of order $k$ at the equilibrium point shows that this is not a point of minimum. The project is focused in the particular case of 4 degrees of freedom where there's a splitting of the potential energy into 2 planes, and we try to understand the situation where we have 2 natural directions to look for an assymptotic orbit to the equilibrium point and that, in the momento, also means there's no known technique to effectively "find'' such trajectory. Results in this direction can also give new and interesting insight into the study of the problem known as the inversion of the Lagrange-Dirichlet theorem. (AU)