Programme

• Some Recent Developments in Symmetries and Conservation Laws for Partial Differential Equations
  George Bluman, University of British Columbia, Vancouver, Canada
  
  Description: This course will be concerned with recent developments on how to find and use symmetries for PDEs; how to find and use conservation laws for PDEs; connections between symmetries and conservation laws for PDEs. Much of the material in this course will appear in a forthcoming book to be published in the Springer Applied Mathematical Sciences series. Background material appears in Symmetry and Integration Methods for Differential Equations (Springer 2002) by George W. Bluman and Stephen C. Anco, which focused on Lie groups of transformations and their applications to solving ordinary differential equations (ODEs), the construction of conservation laws (integrating factors) for ODEs, and finding invariant solutions of PDEs.
  Topics, Overview
  
  
  
• The Inverse Problem in the Calculus of Variations and the Geometry of Second Order ODEs.
  Geoff Prince, La Trobe University, Melbourne, Australia Description: The inverse problem in the calculus of variations is the mathematical generalisation of the problem of deciding whether a given system of ODEs is derivable from a Lagrangian of T-V type. The inverse problem arises in general relativity, Finsler geometry, classical mechanics and elsewhere. Apart from its practical application, it has inspired significant developments in the geometrisation of second order ODEs, in particular many of the theorems of classical mechanics and the geometry of geodesics can now be extended to a far broader class of equations. This course introduces the geometric theory of ODEs due to Klein, Crampin, Sarlet and others and the resulting formulation of the inverse problem. In the second half we examine the exterior differential system approach to the solution of the inverse problem. Some background in exterior calculus and tangent bundle geometry is required.