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26-29 April 2010   master course

Marco Castrillón López, Department of Geometry and Topology, University Complutense of Madrid

Variational calculus and field theoryFacultat de Matemàtiques i Estadística, room 100. 
Schedule: Monday 26 April, 12-14 h and 16-18 h; Wednesday 28 April, 16-19 h; Thursday 29 April, 11-14 h.

Contents   The aim of the course is to introduce and develop the basic tools of geometric variational calculus, with emphasis on field theory, as well as some applications to models of theoretical physics.

  • Fibre bundles. Jet bundles. Examples.
  • Lagrangians and variational principles. Euler-Lagrange equations. Intrinsic formulation of the Poincaré-Cartan form.
  • Classical examples. Klein-Gordon equations. Maxwell equations.
  • Introduction to gauge symmetries. Utiyama theorem and Yang-Mills equations.
  • Finite-dimensional symmetry groups. Euler-Poincaré reduction.

Basic bibliography  

  • G. Giachetta, L. Mangiarotti, G. Sardanashvily, Advanced classical field theory, World Scientific, 2009.
  • K.B. Marathe, G. Martucci, Mathematical foundations of gauge theories, North-Holland, 1992.
  • D.J. Saunders, The geometry of jet bundles, Cambridge University Press, 1989.