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A paradigm for joined Hamiltonian and dissipative systems. (English) Zbl 0661.70025

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.

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
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References:

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