2nd International Summer School on Geometry, Mechanics, and Control

TUDOR RATIU

Section de Mathématiques,  Faculté des Sciences de Base, EPFL,  Lausanne, Switzerland

Symmetry in geometric mechanics and dynamics

Presentation

Material

This course presents the basic notions of geometric mechanics: Hamiltonian systems on symplectic and Poisson manifolds, the momentum map, reduction theory, the slice coordinates, the tangent-normal decomposition of a symmetric system, relative equilibria, and the fundamental notions on linear and nonlinear stability in the context of symmetry.

JAIR KOILLER

Fundação Getulio Vargas, BrazilTopics in vortex dynamics

Topics in vortex dynamics

Presentation

Material

The mathematics of vorticity started with Euler 250 years ago.   We begin with a review of basic hydrodynamics, in the two limits,  Eulerian (no viscosity) x Stokesian (infinite viscosity). In the Eulerian realm, vorticity forms the `sinews and muscles' for fluid motion.  We focus on 2 dimensional flows, where complex variables and Riemann surfaces come to fore.  We will discuss some of the huge amount of classical and recent literature in the area, selecting papers related to the other minicourses.

Towards the end we will present a formulation for vortex motion on surfaces, that we obtained in collaboration with S.Boatto.

JUAN CARLOS MARRERO

Universidad de La Laguna, Spain

Continuous Mechanics and Lie algebroids
 

Presentation                slide1   slide2

Material

The category of Lie algebroids has proves useful to formulate problems in differential fields of mathematics. In the context of Mechanics an ambitions to develop formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids. In this talk I will present some recent ideas in the above direction. In first part, a Klein formalism for Lagrangian Mechanics and a symplectic formulation for Hamiltonian Mechanics (both in the Lie algebroid setting) is developed. The key idea is to use the prolongation of a Lie algebroid over a fibration. In the second part, I will discuss nonholonomic mechanical systems subjected to linear constraints on Lie algebroids. Different examples which illustrate the resulsts will be also presented.

DAVID MARTIN DE DIEGO

ICMAT, CSIC, Spain

Discrete Mechanics and  groupoids

Presentation

Material

In this survey talk, we will give a review of discrete mechanics, which permits the construction of geometric integrators for Lagrangian mechanical systems. Moreover, we extend this construction to reduced Lagrangian systems using the Lie groupoid theory. Therefore, we will obtain geometric integrators for Euler-Poincaré equations, Lagrange-Poincaré equations...with remarkable geometric properties.

Finally, we will extend these constructions to the case of nonholonomic constraints.