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On some apllications of Lie Algebroids in Geometry and Physics

Gheorghiu, Irina Mihaela
Cariñena Marzo, José Fernando (dir.) ; Martínez Fernández, Eduardo (dir.)

(Física Teórica)

Resumen (otro idioma): The main purpose of our work is to present applications of the Lie algebroid structure in both mathematical and physical context. In the first chapter we have introduced the notion of Lie algebroid, presenting a number of examples, and we have presented some useful properties that we used later on. One of our principal results in the mathematical part was to give a generalization of the notion of Jacobi fields corresponding to sode on manifolds and on Lie algebroids. We have done that considering a new take on a first order variational equation on a manifold. We also generalized the Jacobi equation for this generalized cases of Jacobi fields associated to sode. For that we had to generalize the non-linear connection and the Jacobi endomorphism to the context of Lie algebroid. We used this theory in the particular instance of a geodesic spray on a Riemannian Lie algebroid. For this case we have shown that an integral curve of it has no conjugate points along it if and only if it minimizes the energy functional of the system whose solution are given by the geodesic spray. To exemplify the theorem we considered the space of skew-symmetric matrices of dimension 3 who has a Lie algebroid structure. In Chapter 4, for the physical counterpart, we analyzed the virial theorem in the first place for mechanical systems and nonholonomic systems on the tangent bundle, and afterwards, for unconstrained and nonholonomic systems on Lie algebroids. We could prove that a virial like theorem holds for systems on Lie algebroids, fact that will allow us to obtain information about the time average of the action of the dynamical section upon the virial function for more systems than before due to the wide range of systems that can be described with the help of a Lie algebroid structure. Also in this chapter we have presented in detail instances of this theorem through some examples. We find interesting for further investigation to see if the minimizing theorem presented here takes place for any Lagrangian, not necessarily a Riemannian one and for the other topology. Precisely see in what conditions the result holds when we look for the geodesic to be a strong minimum for the energy functional.

Pal. clave: física teórica ; matemáticas

Área de conocimiento: Física teórica

Departamento: Física Teórica

Nota: Tesis-Univ. Zaragoza, Física Teórica, 2016

Registro creado el 2016-03-08, última modificación el 2019-02-19

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