Geodesic Vlasov equations and their integrable moment closures
Darryl D. Holm Cesare Tronci
Journal of Geometric Mechanics 2009, 1(2): 181-208 doi: 10.3934/jgm.2009.1.181
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.
keywords: moment dynamics. Kinetic theory momentum map
Hybrid models for perfect complex fluids with multipolar interactions
Cesare Tronci
Journal of Geometric Mechanics 2012, 4(3): 333-363 doi: 10.3934/jgm.2012.4.333
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.
keywords: Vlasov kinetic theory order parameter Complex fluid Yang-Mills theories.
Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups
François Gay-Balmaz Cesare Tronci Cornelia Vizman
Journal of Geometric Mechanics 2013, 5(1): 39-84 doi: 10.3934/jgm.2013.5.39
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.
keywords: Bundle automorphism chromomorphism group. momentum map
Collisionless kinetic theory of rolling molecules
Darryl D. Holm Vakhtang Putkaradze Cesare Tronci
Kinetic & Related Models 2013, 6(2): 429-458 doi: 10.3934/krm.2013.6.429
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.
keywords: Nonholonomic dynamics collisionless kinetics Euler-Poncaré reduction.

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