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Doctoral Thesis
DOI
10.11606/T.45.2007.tde-03102007-162259
Document
Author
Full name
Ricardo dos Santos Freire Junior
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2007
Supervisor
Committee
Garcia, Manuel Valentim de Pera (President)
Lopes, Orlando Francisco
Savi, Marcelo Amorim
Tal, Fabio Armando
Teixeira, Marco Antonio
Title in Portuguese
Instabilidade de pontos de equilíbrio de alguns sistemas lagrangeanos
Keywords in Portuguese
estabilidade de Liapunov
sistemas lagrangeanos
teorema de Dirichlet-Lagrange
Abstract in Portuguese
Neste trabalho, estudamos algumas inversões parciais do teorema de Dirichlet-Lagrange, essencialmente estendendo os resultados em dois graus de liberdade de Garcia e Tal (2003) para algumas situações em $R^$. Mais precisamente, um dos objetivos é mostrar, no contexto da mecânica lagrangeana, que se há um split da energia potencial em uma parte no plano cujo jato $k$ mostra que ela não tem mínimo no ponto de equilíbrio e existe o jato $k-1$ do seu gradiente, e a outra em $R^$ que tenha mínimo no ponto de equilíbrio, este é instável. A instabilidade do ponto de equilíbrio em estudo é provada mostrando a existência de uma trajetória assintótica ao mesmo. Para isso, apresentamos um resultado inicial para lagrangeanos com uma forma bem específica e, a seguir, mostramos que a classe de lagrangeanos que descrevemos acima pode ser levada a esta forma, através de uma adequada mudança de coordenadas espaciais. Além disso, consideramos a extensão desses resultados a sistemas com forças giroscópicas.
Title in English
Instability of Equilibrium Points of Some Lagrangian Systems
Keywords in English
Lagrange-Dirichlet theorem
lagrangian systems
Liapunov stability
Abstract in English
In this work, we study some partial inversions of the Lagrange-Dirichlet theorem, extending the results in two degrees of freedom of Garcia and Tal (2003) for some other situations in $\mathbb^$. More precisely, one of our objectives is to show, in the context of lagrangian mechanics, that if there is a splitting of the potential energy in one part in the plane which its $k$-jet shows that it does not have a minimum in the equilibrium and there exists the $(k-1)$-jet of its gradient, and the other part in $\mathbb^$ has a minimum in the equilibrium, then the equilibrium point is unstable. Instability of the equilibrium point is shown by proving the existence of an assymptotic trajectory to it. For this purpose, first it is proven a result for lagrangians with a specific form and, next, we show that the class of lagrangians we are interested in can be transformed into this specific form by a subtle change of spatial coordinates. Finally, we consider the extension of this results to systems with gyroscopic forces.
 
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TeseRicardoFreire.pdf (321.20 Kbytes)
Publishing Date
2007-10-15
 
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