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### Open Access Journals

JGM

Various integrable geodesic flows on Lie groups are shown to arise by taking moments of a geodesic Vlasov equation on the group of canonical transformations. This was already known for both the one- and two-component Camassa-Holm systems [18, 19]. The present paper extends our earlier work to recover another integrable system of ODE's that was recently introduced by Bloch and Iserles [5].
Solutions of the Bloch-Iserles system are found to arise from the Klimontovich solution of the geodesic Vlasov equation. These
solutions are shown to form one of the legs of a dual pair of momentum maps.
The Lie-Poisson structures for the dynamics of truncated moment hierarchies are also presented in this context.

JGM

Multipolar order in complex fluids is described by statistical correlations. This paper presents a novel dynamical approach, which accounts for microscopic effects on the order parameter space. Indeed, the order parameter field is replaced by a statistical distribution function that is carried by the fluid flow. Inspired by Doi's model of colloidal suspensions, the present theory is derived from a

*hybrid*moment closure for Yang-Mills Vlasov plasmas. This hybrid formulation is constructed under the assumption that inertial effects dominate over dissipative phenomena (perfect complex fluids), so that the total energy is conserved and the Hamiltonian approach is adopted. After presenting the basic geometric properties of the theory, the effect of Yang-Mills fields is considered and a direct application is presented to magnetized fluids with quadrupolar order (*spin nematic*phases). Hybrid models are also formulated for complex fluids with symmetry breaking. For the special case of liquid crystals, the moment method can be applied to the hybrid formulation to study to the dynamics of cubatic phases.
JGM

We formulate Euler-Poincaré equations on the Lie group
$Aut(P)$ of automorphisms of a principal bundle $P$. The corresponding flows are referred to as EP$Aut$ flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle $P$, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction.
This approach leads naturally to a dual pair structure containing $\delta\text{-like}$ momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff).
In the second part, we consider incompressible flows
on the Lie group $Aut_{vol}(P)$ of volume-preserving bundle
automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.

KRM

We derive a collisionless kinetic theory for an ensemble of molecules undergoing nonholonomic rolling dynamics. We demonstrate that the existence of nonholonomic constraints leads to problems in generalizing the standard methods of statistical physics. In particular, we show that even though the energy of the system is conserved, and the system is closed in the thermodynamic sense, some fundamental features of statistical physics such as invariant measure do not hold for such nonholonomic systems.
Nevertheless, we are able to construct a consistent kinetic theory using Hamilton's variational principle in Lagrangian variables, by regarding the kinetic solution as being concentrated on the constraint distribution. A cold fluid closure for the kinetic system is also presented, along with a particular class of exact solutions of the kinetic equations.

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