Variational Integrators for Euler-Poincaré equations
Presenting author: César Rodrigo
On a principal $ G$-bundle, variational field theories
that are invariant by the action of a certain subgroup $ H\subset G$ lead to associated reduced
theories, where the reduced fields are described by a principal connection and an $ H$-structure, and constrained by the condition
that the connection is flat and the H-structure is parallel with respect to the connection.
For any $ H$-invariant Lagrangian density, fields that satisfy the corresponding constrained (Euler-Poincaré)
variational principle are characterized by Euler-Poincaré equations.
Using simplicial complexes to discretise the base manifold of the principal bundle, and Ehresman's Gauge
groupoid to discretize principal connections, we describe a corresponding bundle of reduced discrete fields.
For any particular lagrangian function on this discrete bundle, we introduce a corresponding
constrained variational principle that represents a discrete analogue of Euler-Poincaré
variational principle. Critical points of this variational principle are characterized by
corresponding discrete Euler-Poincaré equations. We prove a corresponding theorem of existence and uniqueness of solutions,
for fixed values on an initial band, if the reduced discrete Lagrangian satisfies a certain regularity condition.
The proof is constructive, leading to a particular algorithm (integration algorithm), based on the existence of
an inverse of the momentum map associated to the reduced discrete Lagrangian, extending in this way to field theories
recent variational integrators for mechanics on Lie groups.
Relevant Bibliography:
Euler-Poincaré reduction in principal bundles by a subgroup of the structure group. J. Geom. Phys. 74, 352–-369 (2013).
Lagrange-Poincaré field equations, J. Geom. Phys. 61 2120--2146 (2011).
First variation formula and conservation laws in several independent discrete variables. J. Geom. Phys. 62(1), 61--86 (2012).
Euler-Poincaré reduction for discrete field theories. J. Math. Phys. 48(3), 32902--32917 (2007).
Symmetry-preserving discretization of variational field theories. arXiv preprint arXiv:1509.08750 (2015).
A discrete theory of connections on principal bundles. arXiv preprint math/0508338 (2005).
Discrete Lagrangian reduction, discrete Euler–Poincaré equations, and semidirect products.
Letters in Mathematical Physics 49(1) 79--93 (1999).
Click here to access the slides.