Reduction by stages in field theory
Loading...
Download
Official URL
Full text at PDC
Publication date
2024
Defense date
14/04/2023
Authors
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Universidad Complutense de Madrid
Citation
Abstract
La mecánica geométrica investiga cómo las técnicas globales y covariantes de la geometría pueden utilizarse para estudiar una amplia variedad de problemas de la mecánica diferencial clásica que, a primera vista, pueden parecer desconectados. En este paradigma podemos estudiar sólidos rígidos, movimientos planetarios, vehículos rodados, control de enjambres de cuadricópteros y robots esféricos entre muchos otros sistemas. Una de las ideas fundamentales de esta área de investigación afirma que la simetría de los sistemas físicos da lugar con bastante frecuencia a cantidades conservadas y versiones reducidas, potencialmente más sencillas, del problema. Un número considerable de sistemas presentan simetrías con naturaleza notablemente diferenciada e impulsan la tarea de realizar por etapas su proceso de reducción. Primero, se obtiene un sistema reducido utilizando las simetrías de un tipo para, posteriormente, reducir el sistema resultante mediante las simetrías de otra naturaleza. Este proceso iterativo ya ha sido profusamente estudiado en mecánica clásica tanto Lagrangiana como Hamiltoniana...
Geometric mechanics delves on how the global and coordinate-independent techniques ofdifferential geometry can be used to study a wide variety of problems in classical mechanicswhich, at first sight, seem to be unrelated. Rigid bodies, planetary motion, rolling vehicles,control of swarms of quadrotors, sphere robots, among many other systems can be studiedwithin this paradigm.One of the cores ideas of this area of research is that symmetries of the physical systemdo quite often produce conserved quantities and lead to a reduced (and potentially simpler)version of the problem. A fair number of mechanical systems present symmetries with aremarkably distinct nature and prompt the task to carry their reduction procedure by stages.First, a reduced system is obtained using the symmetries of one kind and, afterwards, theresulting system is reduced using the symmetries of the other nature. This iterative processhas already been studied with profusion in both, Lagrangian and Hamiltonian classical mechanics...
Geometric mechanics delves on how the global and coordinate-independent techniques ofdifferential geometry can be used to study a wide variety of problems in classical mechanicswhich, at first sight, seem to be unrelated. Rigid bodies, planetary motion, rolling vehicles,control of swarms of quadrotors, sphere robots, among many other systems can be studiedwithin this paradigm.One of the cores ideas of this area of research is that symmetries of the physical systemdo quite often produce conserved quantities and lead to a reduced (and potentially simpler)version of the problem. A fair number of mechanical systems present symmetries with aremarkably distinct nature and prompt the task to carry their reduction procedure by stages.First, a reduced system is obtained using the symmetries of one kind and, afterwards, theresulting system is reduced using the symmetries of the other nature. This iterative processhas already been studied with profusion in both, Lagrangian and Hamiltonian classical mechanics...
Description
Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 14-04-2023