Uniformization of Nonlinear Hamiltonian Systems of Vlasov and Hartree Type

Details Uniformization of Nonlinear Hamiltonian Systems of Vlasov and Hartree Type Uniformization of Nonlinear Hamiltonian Systems of Vlasov and Hartree Type

2005-12-15

arxiv.org/abs/math-ph/0512051v1

Teoreticheskaya i matematicheskaya Fizika, Vol. 33 (1977), No. 1, pp17--31

Physics

Mathematical Physics

18 pages. This paper is translated from Russian by Plenum Publishing Corporation

Scientific paper

Nonlinear Hamiltonian systems describing the abstract Vlasov and Hartree equations are considered in the framework of algebraic Poissonian theory. The concept of uniformization is introduced; it generalizes the method of second quantization of classical systems to arbitrary Hamiltonian (Lie-Jordan, or Poissonian) algebras, in particular to the algebras of operators in indefinite spaces. A functional calculus is developed for the uniformized observables, which unifies the calculi of generating functionals for classical, Bosonian and Fermionian multiparticle variables. The nonlinear kinetic equation of Vlasov and Hartree type are derived for both classical and quantum multiparticle Hamiltonian systems in the mean field approximation. The possibilities for finding approximate solutions of these equations from the solutions of uniformized linear equations are investigated.

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