Dynamics of a galactic Hamiltonian system

Abstract

Abstract. We study an even polynomial potential which appears in the study of the galactic dynamics. We prove the existence of four families of periodic orbits in every positive energy level, and we compute an analytic approximation of them. Using such periodic orbits we provide information about the non-integrability of this Hamiltonian system.

On the analytic integrability of the cored galactic Hamiltonian

“…The potential √ 1 + x 2 + y 2 q 2 has an absolute minimum and a reflection symmetry with respect the two axis x and y. The motivation for the choice of these symmetries comes from the interest of this potential in galactic dynamics, see for instance [2,3,4,9,10,11]. The parameter q gives the ellipticity of the potential, which ranges in the interval 0.6 ≤ q ≤ 1.…”

On the integrability of a three-dimensional cored galactic Hamiltonian

“…The motivation for the choice of the potential 1 + x 2 + y 2 + z 2 /q comes from the interest of this potential in galactic dynamics, see for instance [2,3,5,6,7,9,12,13,14,15,16]. The parameter q gives the ellipticity of the potential, which ranges in the interval √ 0.6 ≤ q ≤ 1.…”

“…Let U be an open and dense set in R 6 . We say that the non-locally constant function F : U → R is a first integral of the vector field X on U , if F (x(t), y(t), z(t), p x (t), p y (t), p z (t)) = constant for all values of t for which the solution (x(t), y(t), z(t), p x (t), p y (t), p z (t)) of X is defined in U .…”

Sistema de Rössler e dinâmica de galáxias: aplicações do Teorema da Média

“…Usando a Teoria da Média vamos demonstrar a existência de duas famílias deórbitas periódicas destas equações hamiltonianas para ε > 0 suficientemente pequeno. Os resultados a seguir podem ser encontrados em [6]. O sistema hamiltonianoé dado por:…”

“…No quarto capítulo também aplicaremos a Teoria da Média e ainda vamos usar as Funções Elípticas Jacobianas, as quais veremos no primeiro capítulo, para estudar um sistema hamiltoniano introduzido em [6].…”