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Multidimensional Euler-Poincaré equations
No full textProceedings of the 8th International Conference held in Opava, August 27–31, 2001.This work is devoted to presenting a summary of results developed mainly by the first author, T. S. Ratiu and S. Shkoller in their previous work [Proc. Amer. Math. Soc. 128 (2000), no. 7, 2155–2164;].
The main results concern the reduction of a Lagrangian field theory under a group of symmetries, obtaining the analog of the Euler-Poincaré equations, which are also proved to be equivalent to a Noether conservation law given by the symmetry. Furthermore, the compatibility condition needed for obtaining solutions of the original variational problem starting from the solutions of the reduced system is also stated. A final example is given.
The paper is written in geometrical language.Depto. de Álgebra, Geometría y TopologíaFac. de Ciencias MatemáticasTRUEpu
Cartan geometries and their symmetries: a Lie algebroid approach
No full textIn this book we first review the ideas of Lie groupoid and Lie algebroid, and the associated concepts of connection. We next consider Lie groupoids of fibre morphisms of a fibre bundle, and the connections on such groupoids together with their symmetries. We also see how the infinitesimal approach, using Lie algebroids rather than Lie groupoids, and in particular using Lie algebroids of vector fields along the projection of the fibre bundle, may be of benefit. We then introduce Cartan geometries, together with a number of tools we shall use to study them. We take, as particular examples, the four classical types of geometry: affine, projective, Riemannian and conformal geometry. We also see how our approach can start to fit into a more general theory. Finally, we specialize to the geometries (affine and projective) associated with path spaces and geodesics, and consider their symmetries and other properties
Calculus of forms along a map adapted to the study of second-order differential equations
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A geometric theory of ordinary first order variational problems in fibered manifolds. I. Critical sections
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