Journal of Geometric Mechanics
September 2012 , Volume 4 , Issue 3
dedicated to Tudor S. Ratiu on the occasion of his 60th birthday
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This number is the third one that the Journal of Geometric Mechanics dedicates to the works presented on the occasion of the 60th birthday celebration of Tudor S. Ratiu, held at the Centre International de Rencontres Mathématiques de Luminy. As it was the case for the first two numbers (volume 3(4) and volume (4)2), we hope that the articles contained in this special number will give the reader an appreciation for the importance of the contributions of Tudor S. Ratiu to the fields that are at the core of this journal.
A close relationship between the classical Hamilton-Jacobi theory and the kinematic reduction of control systems by decoupling vector fields is shown in this paper. The geometric interpretation of this relationship relies on new mathematical techniques for mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to describe in a simplified way the mechanics of nonholonomic systems with both control and external forces.
We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. First, we decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian vector fields and its complement. The latter is related to dual space of the Lie algebra. We identify generators of homotheties as dynamically irrelevant vector fields in the complement. Lie algebra of Hamiltonian vector fields is isomorphic to the space of all Lagrangian submanifolds with respect to Tulczyjew symplectic structure. This is obtained as tangent space at the identity of the group of canonical diffeomorphisms represented as space of sections of a trivial bundle. We obtain the momentum-Vlasov equations as vertical equivalence, or representative, of complete cotangent lift of Hamiltonian vector field generating particle motion. Vertical representatives can be described by holonomic lift from a Whitney product to a Tulczyjew symplectic space. We show that vertical representatives of complete cotangent lifts form an integrable subbundle of this Tulczyjew space. We exhibit dynamical relations between Lie algebras of Hamiltonian vector fields and of contact vector fields, in particular; infinitesimal quantomorphisms on quantization bundle. Gauge symmetries of particle motion are extended to tensorial objects including complete lift of particle motion. Poisson equation is then obtained as zero value of momentum map for the Hamiltonian action of gauge symmetries for kinematical description.
Given a linear (algebraic) group $G$ acting on real or complex $n$-space, we determine all the common invariant sets of $G$-symmetric vector fields. It turns out that the investigation of certain algebraic varieties is sufficient to characterize these invariant sets forced by symmetry. Toral, compact and reductive groups are discussed in some detail, and examples, including a Couette-Taylor system, are presented.
We show that if a smooth multiplicative subbundle $S\subseteq TG$ on a groupoid $G⇉P$ is involutive and satisfies completeness conditions, then its leaf space $G/S$ inherits a groupoid structure over the space of leaves of $TP\cap S$ in $P$.
As an application, a special class of Dirac groupoids is shown to project by forward Dirac maps to Poisson groupoids.
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.
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