Abstract:
I will present the two competing theories of liquid crystals, widely used today: Ericksen-Leslie nematodynamics and Eringen micropolar theory. Nematodynamics describes the evolution of liquid crystals in the nematic phase, that is, the molecules exhibit a preferred direction. Micropolar theory regards the molecules as elastic small bodies. These two competing sets of equations will be presented only in conservative form in order to understand their geometric underpinning. Methods of geometric mechanics will be used in order to show their Hamiltonian character.
Then, I will sketch the solution of a 15 year old open problem, a statement of Eringen affirming that his theory includes the Ericksen-Leslie theory. This has never been proved to be true and papers attempting to show it are mathematically wrong. Our solution uses the theory of reduction in an essential way and gives a proof of this statement. In the process, two new equations for liquid crystals are developed, one of them generalizing Ericksen-Leslie in such a way that disclinations can be treated, something that the original Ericksen-Leslie equations could not do. The term disclinations appears in the liquid crystal theory and means certain discontinuities of some of the variables.