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Conference: Symposium on Mathematical Physics


Recent progresses in geometric continuum mechanics: liquid crystals and multisymplectic integrators

Dr. François Gay-Balmaz'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 Marsden-Weinstein 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 Ericksen-Leslie 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.