A forma normal de Gustavson e aplicações no estudo da estabilidade de soluções de equilíbrio de sistemas hamiltonianos degenerados

• Autor(a) principal: Nascimento, Jakeline Cleide Santos Almeida do
• Data de Publicação: 2022
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da UFS
• Texto Completo: http://ri.ufs.br/jspui/handle/riufs/17363
• topic: Matemática; Sistema hamiltoniano; Estabilidade; Forma normal de Gustavson; Forma normal de Birkhoff; Hamiltonian system; Stability; Gustavson normal form; Birkhoff normal form; CIENCIAS EXATAS E DA TERRA::MATEMATICA

In this dissertation, we aim to study the Gustavson normal form of a Hamiltonian system in the degenerate and non-degenerate cases and apply the cases in which a Hamiltonian function has one degree of freedom and starts with third or fourth order terms in the study of the stability of solutions of balance. To achieve our goal, we include a preliminary chapter where we provide the basic theory of Hamiltonian systems, transformations symplectics and generating functions, in addition to some basic concepts of the qualitative theory of ordinary differential equations. We approach the classical theory of the form Gustavson and Birkhoff normal in the non-degenerate cases where the matrix of the linearized system is diagonalizable and the pure imaginary eigenvalues, in addition In addition, we provide some recent results in the case of systems with a degree of of freedom in which the Hamiltonian function is represented as a series of powers starting with cubic or fourth-order terms. In both cases, the main technique is to obtain a symplectic transformation given by a generating function in order to simplify the structure of higher-order terms of the Hamiltonian function. In the end, we will use the normal forms to obtain some conditions for the stability of equilibrium solutions in the case of systems degenerates with one degree of freedom.