A paradigm for joined Hamiltonian and dissipative systems

Morrison, Philip

doi:10.1016/0167-2789(86)90209-5

Cited by 204 publications

(269 citation statements)

References 14 publications

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“…This is readily proved (see [20]) by realizing that, thanks to (14), Q is a Lyapunov quantity, so that points x 0 such that ∂ i Q (x 0 ) = 0 are places towards which the motion (13) will converge. If G were a semi-definite negative tensor, equally some Q' could be defined to play the opposite role, i.e., that of a monotonically decreasing quantity, and the asymptotically stable equilibrium points would then be its minima.…”

Section: Dissipative Systems and Metric Algebramentioning

confidence: 99%

“…In [20] a complete system is referred to as a system that conserves its energy, but redistributes it in an irreversible way: this "irreversible redistribution" is named dissipation. Complete systems described via an MBA are indicated as complete metriplectic systems (CMS).…”

Section: Energy Conservation Entropy Increasementioning

confidence: 99%

“…The bracket (., . ), referred to as (semi)metric bracket, is a type of Leibniz bracket quite different from Poisson algebra [6,20]. Once a generating function Q (x) is defined so that the dynamics of the system is governed by…”

Section: Dissipative Systems and Metric Algebramentioning

confidence: 99%

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Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Materassi

2016

Entropy

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This lecture is a short review on the role entropy plays in those classical dissipative systems whose equations of motion may be expressed via a Leibniz Bracket Algebra (LBA). This means that the time derivative of any physical observable f of the system is calculated by putting this f in a "bracket" together with a "special observable" F, referred to as a Leibniz generator of the dynamics. While conservative dynamics is given an LBA formulation in the Hamiltonian framework, so that F is the Hamiltonian H of the system that generates the motion via classical Poisson brackets or quantum commutation brackets, an LBA formulation can be given to classical dissipative dynamics through the Metriplectic Bracket Algebra (MBA): the conservative component of the dynamics is still generated via Poisson algebra by the total energy H, while S, the entropy of the degrees of freedom statistically encoded in friction, generates dissipation via a metric bracket. The motivation of expressing through a bracket algebra and a motion-generating function F is to endow the theory of the system at hand with all the powerful machinery of Hamiltonian systems in terms of symmetries that become evident and readable. Here a (necessarily partial) overview of the types of systems subject to MBA formulation is presented, and the physical meaning of the quantity S involved in each is discussed. Here the aim is to review the different MBAs for isolated systems in a synoptic way. At the end of this collection of examples, the fact that dissipative dynamics may be constructed also in the absence of friction with microscopic degrees of freedom is stressed. This reasoning is a hint to introduce dissipation at a more fundamental level.

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Section: Dissipative Systems and Metric Algebramentioning

confidence: 99%

Section: Energy Conservation Entropy Increasementioning

confidence: 99%

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Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Materassi

2016

Entropy

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“…However, due to the fact that the Jacobi identity is not satisfied, there is some dissipation associated with the almost-Poisson systems, which has been coined "fake dissipation" in Ref. 2 . This fake dissipation enters in competition with the dissipative terms introduced in the equations of motion.…”

mentioning

confidence: 99%

“…However here the convergence towards attractors is very slow, which explains why the trajectories appear to fill densely some parts of phase space. We compared the dissipative dynamics (ε = 0) to another type of dissipation, a metriplectic system 2,5,6 . The idea is to construct a conservative system with an entropy.…”

mentioning

confidence: 99%

Conservative dissipation: How important is the Jacobi identity in the dynamics?

Caligan¹,

Chandre²

2016

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Hamiltonian dynamics are characterized by a function, called the Hamiltonian, and a Poisson bracket. The Hamiltonian is a conserved quantity due to the anti-symmetry of the Poisson bracket. The Poisson bracket satisfies the Jacobi identity which is usually more intricate and more complex to comprehend than the conservation of the Hamiltonian. Here we investigate the importance of the Jacobi identity in the dynamics by considering three different types of conservative flows in R 3 : Hamiltonian, almost-Poisson and metriplectic. The comparison of their dynamics reveals the importance of the Jacobi identity in structuring the resulting phase space.

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Non‐Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

Kobelev

2019

Comp and Math Methods

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The non‐Leibniz Hamiltonian and Lagrangian formalism for the certain class of dissipative systems is introduced in this article. The formalism is based on the generalized differentiation operator (κ‐operator) with a nonzero Leibniz defect. The Leibniz defect of the introduced operator linearly depends on one scaling parameter. In a special case, if the Leibniz defect vanishes, the generalized differentiation operator reduces to the common differentiation operator. The κ‐operator allows the formulation of the variational principles and corresponding equations of Lagrange and Hamiltonian types for dissipative systems. The solutions of some generalized dynamical equations are provided closed form.

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A paradigm for joined Hamiltonian and dissipative systems

Cited by 204 publications

References 14 publications

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Conservative dissipation: How important is the Jacobi identity in the dynamics?

Non‐Leibniz Hamiltonian and Lagrangian formalisms for certain class of dissipative systems

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