Morrison, Philip J. A paradigm for joined Hamiltonian and dissipative systems. (English) Zbl 0661.70025 Physica D 18, 410-419 (1986). A paradigm for describing dynamical systems that have both Hamiltonian and dissipative parts is presented. Features of generalized Hamiltonian systems and metric sytems are combined to produce what are called metriplectic systems. The phase space for metriplectic systems is equipped with a bracket operator that has an antisymmetric Poisson bracket part and a symmetric dissipative part. Flows are obtained by means of this bracket together with a quantity called the generalized free energy, which is composed of an energy and a generalized entropy. The generalized entropy is some function of the Casimir invariants of the Poisson bracket. Two examples are considered: (1) a relaxing free rigid body and (2) a plasma collision operator that can be tailored so that the equilibrium state is an arbitrary monotonic function of the energy. Cited in 62 Documents MSC: 70H05 Hamilton’s equations 70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 76A02 Foundations of fluid mechanics 76X05 Ionized gas flow in electromagnetic fields; plasmic flow Keywords:dynamical systems; generalized Hamiltonian systems; metriplectic systems; phase space for metriplectic systems; antisymmetric Poisson bracket part PDFBibTeX XMLCite \textit{P. J. Morrison}, Physica D 18, 410--419 (1986; Zbl 0661.70025) Full Text: DOI References: [1] Morrison, P. J.; Greene, J. M., Phys. Rev. Lett., 45, 790 (1980) [2] Gardner, C. S., J. Math. Phys., 12, 1548 (1971) [3] Weinstein, A., J. Diff. Geom., 18, 523 (1983) [4] Sudarshan, E. C.G.; Mukunda, N., Classical Mechanics- a Modern Perspective (1983), Krieger: Krieger New York · Zbl 0329.70001 [5] Littlejohn, R., Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, (AIP Conf. Proc., 88 (1982)), 47 [6] Hirsch, M.; Smale, S., Differential Equations, Dynamical Systems and Linear Algebra (1974), Academic Press: Academic Press New York · Zbl 0309.34001 [7] Jordan, D.; Smith, P., Nonlinear Ordinary Differential Equations (1977), Clarendon: Clarendon Oxford · Zbl 0417.34002 [8] Hazeltine, R. D.; Holm, D.; Marsden, J. E.; Morrison, P. J., (ICPP Proc.. ICPP Proc., Lausanne (1984)) [9] Holm, D.; Marsden, J.; Ratiu, T.; Weinstein, A., Physics Reports, 123, 1 (1985) [10] Morrison, P. J.; Hazeltine, R. D., Physics of Fluids, 27, 886 (1984) · Zbl 0585.76175 [11] Center for Pure and Applied Mathematics Report PAM-228 (1984), U.C: U.C Berkeley [12] Physica D (1985), submitted to [13] Kaufman, A. N., Phys. Lett., 109A, 87 (1985) [14] Salmon, R., Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, (AIP Conf. Proc., 88 (1982)), 127 [15] Calkin, M. G., Can. J. Phys., 41, 2241 (1963) [16] Marsden, J. E.; Weinstein, A.; Ratiu, T.; Schmid, R.; Spencer, R. G., Modern Developments in Analytical Mechanics, (IUTAM-ISIMM Symposium (1982)) [17] Morrison, P. J., Mathematical Methods in Hydrodynamics and Integrability in Dynamical Systems, (AIP Conf. Proc., 88 (1982)), 13 [18] Morrison, P. J., Phys. Lett., 80A, 383 (1980) [19] Lenard, A., Ann. of Phys., 3, 390 (1960) [20] Kadomtsev, B. B.; Pogutse, O. P., Phys. Rev. Lett., 25, 1155 (1970) [21] Lynden-Bell, D., Mon. Not. Roy. Astron. Soc., 136, 101 (1967) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.