Explorando Sistemas Hamiltonianos II: A Natureza de Pontos de Equilibrio Degenerados

TL;DR

This paper introduces the blow-up method to analyze the nature of degenerate equilibrium points in Hamiltonian systems, applying it to simple and double pendulum models under constant external torques.

Contribution

It presents the application of the blow-up method to determine the nature of degenerate fixed points in Hamiltonian systems with practical pendulum models.

Findings

Degenerate equilibrium points occur at specific torque values.

The blow-up method effectively classifies fixed points.

Applications to simple and double pendulums demonstrate the method's utility.

Abstract

Neste segundo artigo sobre sistemas Hamiltonianos, apresentamos o metodo da explosao para a determinacao da natureza de pontos fixos (pontos de equilibrio degenerados. Aplicamos o metodo a dois modelos hamiltonianos com um e dois graus de liberdade, respectivamente. Primeiramente, analisamos um sistema formado por um pendulo simples submetido a um torque externo constante T. Em seguida, consideramos um sistema formado por um pendulo duplo com segmentos de comprimentos e massas iguais, tambem submetidos a torques externos constantes e nao nulos. A presenca de pontos de equilibrio degenerados nos casos dos pendulos simples e duplo ocorre para certos valores dos torques externos. In this second article on Hamiltonian systems, we present the blow-up method for the determination of the nature of degenerate fixed points (equilibrium points). We apply the method to two hamiltonian models with one and two degrees of freedom, respectively. Firstly we study a system formed by a simple pendulum submitted to a constant external torque T. Then we consider a system formed by a double pendulum of segments with equal lengths and masses, also submitted to non-vanishing constant external torques. The presence of degenerate equilibrium points in both cases of simple and double pendulums occurs for some values of the external torques.