Kinematical symmetries of three-dimensional incompressible flows

Gümral, Hasan

doi:10.1016/s0167-2789(99)00122-0

Cited by 3 publications

(5 citation statements)

References 51 publications

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“…This formal symplectic set-up has been used to construct Lagrangian and Eulerian invariants of hydrodynamic flows as well as to elucidate the connection between them. The following results has been obtained in [20], [21].…”

Section: Proposition 4 For the Poisson Bi-vector (6) The Bracketsmentioning

confidence: 74%

A time-extended Hamiltonian formalism

Gümral

1999

Physics Letters A

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A Poisson structure on the time-extended space I × M is shown to be appropriate for a Hamiltonian formalism in which time is no more a privileged variable and no a priori geometry is assumed on the space M of motions. Possible geometries induced on the spatial domain M are investigated. An abstract representation space for sl(2, R) algebra with a concrete physical realization by the Darboux-Halphen system is considered for demonstration. The Poisson bi-vector on I ×M is shown to possess two intrinsic infinitesimal automorphisms one of which is known as the modular or curl vector field. Anchored to these two, an infinite hierarchy of automorphisms can be generated. Implications on the symmetry structure of Hamiltonian dynamical systems are discussed. As a generalization of the isomorphism between contact flows and their symplectifications, the relation between Hamiltonian flows on I × M and infinitesimal motions on M preserving a geometric structure therein is demonstrated for volume preserving diffeomorphisms in connection with three-dimensional motion of an incompressible fluid.

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Section: Proposition 4 For the Poisson Bi-vector (6) The Bracketsmentioning

confidence: 74%

A time-extended Hamiltonian formalism

Gümral

1999

Physics Letters A

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“…[21], we introduced, for incompressible flows on a three-dimensional region M of Euclidean space, a symplectic structure on R × M. Using the automorphism algebra of this structure we obtained, in Ref. [22], the generators of volume preserving diffeomorphisms on M and showed that they can, as in two dimensions, be represented by Hamiltonian vector fields. This enabled us to express the Lie-Poisson bracket through the Poisson bracket of invariant functions on M. In this work, we shall utilize these results, which will be summarized in the next section, to construct infinite families of helicity invariants and to obtain a kinematical interpretation of them in the framework of particle relabelling symmetries.…”

Section: Content Of the Workmentioning

confidence: 99%

“…The dynamical formulation on T * Dif f vol (M) when reduced by these symmetries results in the (+)LiePoisson structure on X * div (M). The Eulerian dynamics on the coadjoint orbits is determined by a right-invariant functional on I × M. The Eulerian dynamical equations can be used to construct a formal symplectic structure for (8) on a time-extended domain I × M [21], [22].…”

Section: Kinematical Description and Symplectic Structurementioning

confidence: 99%

“…[22] we shall show that the symplectic set-up of proposition (2) for three dimensional flows is an appropriate modifications of natural geometric tools of two dimensional flows in the sense that it enables us to construct the generators of volume preserving diffeomorphisms and to represent them by Hamiltonian vector fields on M.…”

Section: Propositionmentioning

confidence: 99%

“…(7) define a section of the first jet bundle over I ×M represented by ∂ t +v. As for any such section, this induces the unique connection Γ ≡ dt ⊗ (∂ t + v) on I × M → I [22], [28]- [30]. The connection Γ on I × M dictated by the velocity field v splits the vector fields U k of the form ξ k ∂ t + u k into horizontal and vertical generators…”

Section: Reparametrization and Particle Relabelling Symmetriesmentioning

confidence: 99%

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Helicity invariants in 3D: kinematical aspects

Gümral

2000

Physica D: Nonlinear Phenomena

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Exact, degenerate two-forms Θ k = dθ k on time-extended space R × M which are invariant under the unsteady, incompressible fluid motion on threedimensional region M are introduced. The equivalence class up to exact one-forms of θ k is splitted by the velocity field. The components of this splitting corresponds to Lagrangian and Eulerian conservation laws for helicity densities. These are expressed as the closure of three-forms θ k ∧ Θ l which depend on two discrete and a continuous parameter. Each Θ k is extended to a symplectic form on R × M. The subclasses of θ k 's giving rise to Eulerian helicity conservations is shown to result in conformally symplectic structures on R × M. The connection between Lagrangian and Eulerian conservation laws for helicity is shown to be the same as the conformal equivalence of a Poisson bracket algebra to infinitely many local Lie algebra of functions on R × M.

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Variational aspect and kinetic theory of locally conformal dynamics

Esen,

Gezici,

Gümral

2024

J. Phys. A: Math. Theor.

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We present the locally conformal generalization of the Euler-Lagrange equations. We determine the dual space of the LCS Hamiltonian vector fields. Within this dual space, we formulate the Lie-Poisson equation that governs the kinetic motion of Hamiltonian systems in the context of local conformality. By expressing the Lie-Poisson dynamics in terms of density functions, we derive locally conformal Vlasov dynamics. In addition, we outline a geometric pathway that connects LCS Hamiltonian particle motion to locally conformal kinetic motion.

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Kinematical symmetries of three-dimensional incompressible flows

Cited by 3 publications

References 51 publications

A time-extended Hamiltonian formalism

A time-extended Hamiltonian formalism

Helicity invariants in 3D: kinematical aspects

Variational aspect and kinetic theory of locally conformal dynamics

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