Lagrangian reduction by stages is used to derive the Euler-Poincare equations for the nondissipative coupled motion and micromotion of complex fluids. We mainly treat perfect complex fluids (PCFs) whose order parameters are continuous material variables. These order parameters may be regarded geometrically either as objects in a vector space, or as coset spaces of Lie symmetry groups with respect to subgroups that leave these objects invariant. Examples include liquid crystals, superfluids, Yang-Mills magnetofluids and spin-glasses. A Lie-Poisson Hamiltonian formulation of the dynamics for perfect complex fluids is obtained by Legendre transforming the Euler-Poincare formulation. These dynamics are also derived by using the Clebsch approach. In the Hamiltonian and Lagrangian formulations of perfect complex fluid dynamics Lie algebras containing two-cocycles arise as a characteristic feature.

Euler-Poincare Dynamics of Perfect Complex Fluids

We consider a general approach for the process of Lagrangian and Hamiltonian reduction by symmetries in chiral gauge models. This approach is used to show the complete integrability of several one dimensional texture equations arising in liquid Helium phases and neutron stars.

Lagrangian reductions and integrable systems in condensed matter

https://hal.archives-ouvertes.fr/hal-01396227/document

Lagrangian Reductions and Integrable Systems in Condensed Matter

We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler-Kostant-Symes sense) are naturally built through collective dynamics. In doing so, we address other issues as factorization, Poisson-Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems.

Integrable systems on semidirect product Lie groups

We study integrable systems on the semidirect product of a Lie group and its Lie algebra as the representation space of the adjoint action. Regarding the tangent bundle of a Lie group as phase space endowed with this semidirect product Lie group structure, we construct a class of symplectic submanifolds equipped with a Dirac bracket on which integrable systems (in the Adler–Kostant–Symes sense) are naturally built through collective dynamics. In doing so, we address other issues such as factorization, Poisson–Lie structures and dressing actions. We show that the procedure becomes recursive for some particular Hamilton functions, giving rise to a tower of nested integrable systems.

The Ginzburg-Landau equations for planar textures of superfluid3He are proved to be equivalent to a completely integrable Hamiltonian system. General solutions to these equations are obtained by means of hyperelliptic integrals.

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V. L. Golo

Papers

Solutions to the Ginzburg-Landau equations for planar textures in superfluid 3 He