Reduction and discretization of variational field theories

César Rodrigo Fernández,  Academia Militar  and CINAMIL, CMAF-CIO. 

Data: 9/4/2018

Hora: 15h30

Local: Sala de Seminários- Edifício VII, FCT-UNL

 Abstract:

In this seminar we shall consider smooth and discrete variational field equations on a principal G-bundle, when the Lagrangian admits a subgroup H\subset G of symmetries, and possibly a group of gauge symmetries. This situation appears, for example, in the dynamics of elastic bodies, which will serve to illustrate the contents of the seminar.Solutions of variational (Euler-Lagrange) field equations have Noether current conservation (both for H-symmetries and gauge symmetries), for example work-energy equilibrium in elastic body dynamics. Reducing the variational principle by the subgroup H, the reduced field is represented by a principal connection and an H-structure. Solutions of the reduced variational principle are characterized by Euler-Poincaré equations, and the original gauge symmetries lead to Noether current conservation laws, in H-reduced coordinates. All this holds in the smooth and the discrete formulations.General numerical integrators applied to smooth Euler-Poincaré equations may have a bad performance in Noether current conservation. However discrete Euler-Poincaré equations admit an integration algorithm, that generates an Euler-Poincaré solution for given initial conditions, conserving all discrete Noether currents. A geometrical tool that generates a discrete lagrangian from a smooth one, preserving symmetries, combined with this algorithm, leads then to a numerical scheme (variational integrator) to approximate solutions of smooth Euler-Poincaré equations, with good Noether current conservation properties.