
Conference: Symposium on Mathematical Physics
Recent progresses in geometric continuum mechanics: liquid crystals and multisymplectic integrators
Dr. François GayBalmaz's talk Date: 11.11.14 Time: 14.00  15.00 Room: Y27H28
Symmetry is perhaps one of the most important concept in physics. Its main
uses are the derivation of conservation laws and a possible reduction of
the number of variables needed to describe the system.
In the context of classical mechanics, this is understood via the various
processes of reduction by symmetry, such as MarsdenWeinstein symplectic
reduction. This approach is particularly powerful in the infinite
dimensional context, which covers for example the case of evolutionary
PDEs in continuum mechanics (fluids and nonlinear elasticity).
In this talk we will present two recent progresses made in continuum
mechanics via the tools of reduction by symmetry.
The first concerns the solution of an open problem in liquid crystal
theory. Two competing descriptions (PDEs) of nematic liquid crystal
dynamics have been proposed: the EricksenLeslie director theory and the
Eringen micropolar approach. Up to this day, these two descriptions have
remained distinct in spite of several attempts to show that the micropolar
theory comprises the director theory. We will give a definitive answer by
using Lie group symmetry reduction techniques on nonabelian extensions of
diffeomorphism groups.
The second concerns the development of structure preserving numerical
schemes for PDEs arising in geometrically exact models in nonlinear
elasticity. Such models have common geometric aspects with both liquid
crystals and chiral models. We will present a class of multisymplectic
variational integrators based on a spacetime discretization of the
Hamilton principle for field theories on Lie groups. The resulting scheme
verifies a discrete Noether theorem and is symplectic in both the
temporal and spatial directions.

