Point vortices on the hyperbolic plane

We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. The relative equilibrium conditions are found and the trajectories of relative equilibria with non-zero momentum value are described. We also provide the classification of relative equilibria and the stability criteria for a number of cases, focusing on N=2, 3. Contrary to the system on the sphere, relative equilibria with non-compact momentum isotropy subgroup are found, and are used to illustrate the different stability types of relative equilibria.Comment: To appear in J. Mathematical Physic

Estudio Numérico del Péndulo Elástico de Resorte de Masa No Despreciable.

El propósito de este trabajo es el presentar un estudio numérico del péndulo elástico de resorte de masa no despreciable para pequeñas y grandes deformaciones. Se deriva la condición que ocasiona el fenómeno de resonancia autoparamétrica para pequeñas deformaciones en este sistema. Además, se muestra que las ecuaciones de movimiento de este sistema permiten recuperar el movimiento del péndulo simple y del péndulo elástico en el que el resorte es de masa despreciable para cualquier tipo de deformación. La dinámica de este trabajo consiste en analizar primero el sistema del péndulo simple, posteriormente el del péndulo elástico de resorte de masa despreciable, y finalmente el péndulo elástico de resorte de masa no despreciable

Point vortices on the hyperboloid

In Hamiltonian systems with symmetry, many previous studies have centred their attention on compact symmetry groups, but relatively little is known about the effects of noncompact groups. This thesis investigates the properties of the system N point vortices on the hyperbolic plane H2, which has noncompact symmetry SL(2;R). The Poisson Hamiltonian structure of this dynamical system is presented and relative equilibria conditions are found. We also describe the trajectories of equilibria with momentum value not equal to zero. Finally, stability criteria are found for a number of cases, focusing on N = 2 and 3. These results are placed in with the study of point vortices on the sphere, which has compact symmetry