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Mathematical aspects of quantum fluids. III. Interior Clebsch representations and transformations of symplectic two-cocycles for 4He. (English) Zbl 0635.76135

Summary: [For the former parts see the author, ibid. 26, 2754-2758 (1985; Zbl 0585.76193) and ibid. 27, 2437-2444 (1986; Zbl 0626.76128).]
The symplectic two-cocycle on the semidirect product Lie algebra \(g(\times (W\oplus V\) *\(\oplus V)\) is shown to be canonically related to the dual spaces of the Lie algebras (a) \(g(\times (W\oplus (g(\times V))\) and (b) \(g(\times (W\oplus (g(\times V\) *)). This fact (a) explains the second Poisson bracket for irrotational 4He and (b) leads to a derivation of a new nonlinear Poisson bracket for rotating 4He.

MSC:

76Y05 Quantum hydrodynamics and relativistic hydrodynamics
22E70 Applications of Lie groups to the sciences; explicit representations
76T99 Multiphase and multicomponent flows
81V99 Applications of quantum theory to specific physical systems
70H05 Hamilton’s equations
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References:

[1] Lebedev V. V., Sov. Phys. JETP 48 pp 1167– (1978)
[2] DOI: 10.1016/0003-4916(80)90119-0 · doi:10.1016/0003-4916(80)90119-0
[3] DOI: 10.1016/0375-9601(82)90740-X · doi:10.1016/0375-9601(82)90740-X
[4] DOI: 10.1063/1.526747 · Zbl 0585.76193 · doi:10.1063/1.526747
[5] DOI: 10.1063/1.526983 · Zbl 0626.76128 · doi:10.1063/1.526983
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