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İndeks: Preprint Citation Index × Konu: Lie-Poisson equation ×

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  • Küçük Resim
    Yayın
    Matched pair analysis of the Vlasov plasma
    (Cornell Univ, 2021-02-09) Esen, Oğul; Sütlü, Serkan
    We present the Hamiltonian (Lie-Poisson) analysis of the Vlasov plasma, and the dynamics of its kinetic moments, from the matched pair decomposition point of view. We express these (Lie-Poisson) systems as couplings of mutually interacting (Lie-Poisson) subdynamics. The mutual interaction is beyond the well-known semi-direct product theory. Accordingly, as the geometric framework of the present discussion, we address the matched pair Lie-Poisson formulation allowing mutual interactions. Moreover, both for the kinetic moments and the Vlasov plasma cases, we observe that one of the constitutive subdynamics is the compressible isentropic fluid flow, and the other is the dynamics of the kinetic moments of order > 2. In this regard, the algebraic/geometric (matched pair) decomposition that we offer, is in perfect harmony with the physical intuition. To complete the discussion, we present a momentum formulation of the Vlasov plasma, along with its matched pair decomposition.
  • Küçük Resim
    Yayın
    On extensions, Lie-Poisson systems, and dissipation
    (Cornell Univ, 2021-01-08) Esen, Oğul; Özcan, Gökhan; Sütlü, Serkan
    On the dual space of extended structure, equations governing the collective motion of two mutually interacting Lie-Poisson systems are derived. By including a twisted 2-cocycle term, this novel construction is providing the most general realization of (de)coupling of Lie-Poisson systems. A double cross sum (matched pair) of 2-cocycle extensions are constructed. The conditions are determined for this double cross sum to be a 2-cocycle extension by itself. On the dual spaces, Lie-Poisson equations are computed. We complement the discussion by presenting a double cross sum of some symmetric brackets, such as double bracket, Cartan-Killing bracket, Casimir dissipation bracket, and Hamilton dissipation bracket. Accordingly, the collective motion of two mutually interacting irreversible dynamics, as well as mutually interacting metriplectic flows, are obtained. The theoretical results are illustrated in three examples. As an infinite-dimensional physical model, decompositions of the BBGKY hierarchy are presented. As finite-dimensional examples, the coupling of two Heisenberg algebras and coupling of two copies of 3D dynamics are studied.