A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Prykarpatsky, Anatoliy K.; Bogoli︠u︡bov, N. N.; Golenia, Jolanta

doi:10.1007/s10773-007-9636-3

Cited by 4 publications

(26 citation statements)

References 12 publications

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“…, coinciding with the MHD invariant, presented before in [21,43]. If the above temperature condition does not hold, the equality (87) reduces to the differential relationship…”

Section: As a Simple Example One Can Put αmentioning

confidence: 70%

“…Taking into account that the initial superfluid configuration η 0 ∈ Di f f (M) is fixed, one can define, following reasonings from [43], a new differential relationship…”

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

“…As examples, let us take, following [21,43], the vector field u t := ρ −1 t B t ∈ Γ(T(M)), commuting to the vector field v t ∈ Γ(T(M)), and…”

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

“…following from the net electric fieldẼ t = 0 superconductivity condition (50) along the boundary ∂S t , where S t := η t (S 0 ) ⊂ M is the surface, generated by the diffeomorphism subgroup η t ∈ Di f f (M), t ∈ R, and an arbitrarily chosen surface S 0 = S t | t=0 ⊂ M. The latter is, evidently, equivalent to the equality L d/dt A t |dx t = 0 modulo the gauge transformation A t → A t + ∇ξ t , where L d/dt ξ t + A t |v t = 0 for some function ξ t ∈ C ∞ (M; R) and all t ∈ R. Thus, one can formulate [21,43] the following proposition.…”

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

“…To construct other MHD invariants, depending on the superfluid velocity v t ∈ Γ(T(M)), t ∈ R, let us consider, following [43], two differential one-forms α t |dx t , β t |dx t ∈ Λ 1 (M), x t := η t (X), X ∈ M, satisfying for all t ∈ R the following identity:…”

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

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Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Balinsky

Blackmore

Kycia

et al. 2020

Preprint

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We review a modern differential geometric description of the fluid isotropic motion and featuring it the diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. There is analyzed the adiabatic liquid dynamics, within which, following the general approach, there is explained in detail, the nature of the related Poissonian structure on the fluid motion phase space, as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product. We also present a modification of the Hamiltonian analysis in case of the isotermal liquid dynamics. We study the differential-geometric structure of the adiabatic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and invariant theory. In particular, we construct an infinite hierarchies of different kinds of integral magneto-hydrodynamic invariants, generalizing those before constructed in the literature, and analyze their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, some generalization of the canonical Lie-Poisson type bracket is obtained.

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“…, coinciding with the MHD invariant, presented before in [21,43]. If the above temperature condition does not hold, the equality (87) reduces to the differential relationship…”

Section: As a Simple Example One Can Put αmentioning

confidence: 70%

“…Taking into account that the initial superfluid configuration η 0 ∈ Di f f (M) is fixed, one can define, following reasonings from [43], a new differential relationship…”

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

“…As examples, let us take, following [21,43], the vector field u t := ρ −1 t B t ∈ Γ(T(M)), commuting to the vector field v t ∈ Γ(T(M)), and…”

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

Section: Magneto-hydrodynamic Invariants and Their Geometrymentioning

confidence: 99%

See 3 more Smart Citations

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Balinsky

Blackmore

Kycia

et al. 2020

Preprint

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Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Balinsky

Blackmore

Kycia

et al. 2020

Entropy

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We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented.

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Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

Prykarpatski

2022

Universe

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This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

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A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Cited by 4 publications

References 12 publications

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

Quantum Current Algebra in Action: Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems

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