Tesis doctoral de Ruben Martin Grillo
This dissertation is devoted to the study of geometric formulations of implicit dynamical systems, and more particularly those that are affine in the highest-order derivative. In the first-order case, these implicit dynamical systems are described by an equation of the form ( ,) atx í–x& =b(t,x), where a is a generically singular matrix and b a vector. The main motivation to study this class of systems is that several formalisms of mechanics lead to equations of this kind, amongst them, the most important two: the lagrangian and hamiltonian formalisms. a geometric structure to model these implicit dynamical systems is called linearly singular system and has been previously studied for the time-independent case. An implicit dynamical system on a manifold m is modelled by a vector bundle morphism between the tangent bundle tm and another vector bundle e , together with a section of e . It has been shown how various formalisms of time-independent mechanics are included in this framework. As well, procedures (the so-called constraint algorithms) to find the solutions of a linearly singular system has been developed. In this dissertation, these concepts and results of time-independent linearly singular systems are reviewed and the geometric framework is extended in two directions. first, nonholonomic mechanical systems are formulated and studied in terms of linearly singular systems. To perform this, the concepts of subsystem and quotient of a linearly singular system are introduced. Aspects on regularity, consistency and equations of motion are considered. Symmetries and constants of motion of noholonomic systems are also studied. Implicit hamiltonian systems are also included in this framework. second, a time-dependent version of the linearly singular systems is given and studied. The main difference with respect to the time-independent case is that affine bundles replace vector bundles, and, instead of the tangent bundle , an appropriate jet bundle is used. Specia
Datos académicos de la tesis doctoral «Some geometric aspects of implicit dynamical systems«
- Título de la tesis: Some geometric aspects of implicit dynamical systems
- Autor: Ruben Martin Grillo
- Universidad: Politécnica de catalunya
- Fecha de lectura de la tesis: 22/02/2007
Dirección y tribunal
- Director de la tesis
- Gracia Sabate Francesc Xavier
- Tribunal
- Presidente del tribunal: Miguel Carlos Muñoz lecanda
- eduardo Martinez fernandez (vocal)
- carles Curras bosch (vocal)
- Manuel De león rodríguez (vocal)