Symmetry reduction and unreduction in mechanics and geometry.

`Geometric mechanics' usually stands for the application of differential geometric methods in the study of dynamical systems appearing in mathematical physics. The objectives in this project are all centered around such geometric techniques for reduction and unreduction of a Lagrangian system that is invariant under the action of a symmetry Lie group. One encounters such systems in the context of the calculus of variations and in Finsler geometry. On a principal fibre bundle, the terminology symmetry reduction refers to the fact that an invariant Lagrangian system on the full manifold can be reduced to a system of differential equations on the quotient manifold (the so-called Lagrange-Poincare equations). Unreduction, on the other hand, has the opposite goal: to relate a Lagrangian system on the quotient manifold to a system of differential equations on the full manifold. In this proposal we will investigate the conditions under which the unreduced system can be brought back in the form of a set of Euler-Lagrange equations, for some (yet unknown) Lagrangian on the full manifold. The main tool will be the so-called Inverse Problem of the Calculus of Variations. Besides, we will both extend and specify the method of unreduction in such a way that it fits the needs of the research on isometric submersions between two Finsler manifolds.

Keywords:DIFFERENTIAL EQUATIONS, SYMMETRY, DIFFERENTIAL GEOMETRY, GEOMETRIC MECHANICS

Disciplines:Calculus of variations and optimal control, optimisation, Ordinary differential equations, Mechanics of particles and systems, Differential geometry, Global analysis, analysis on manifolds, Classical mechanics, Classical and quantum integrable systems, Geometric aspects of physics