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Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation

Massimo Materassi
2015 Entropy  
In the present paper, the metriplectic framework for viscous fluids is re-written in the Lagrangian Formulation, where the system is described through material variables: this is a way to describe the  ...  Accordingly, the full metriplectic algebra is constructed in material variables, and this will render it possible to apply it in all those cases in which the Lagrangian Formulation is preferred.  ...  In the present paper, the Lagrangian Formulation (LF) of the metriplectic algebra for a viscous fluid is constructed.  ... 
doi:10.3390/e17031329 fatcat:hxqm6z4zhfehvetjxfw5irg62q

Lagrangian Hydrodynamics, Entropy and Dissipation [chapter]

Massimo Materassi
2015 Hydrodynamics - Concepts and Experiments  
Lagrangian Hydrodynamics, Entropy and Dissipation http://dx.doi.org/10.5772/59319 119 Hydrodynamics -Concepts and Experiments 120 The determinant of the Jacobian is indicated as det .  ...  The description of such modes of the continuum is traditionally and better done in the Eulerian Formalism, where the physical quantities locally describing the motion of a fluid are conceived as local  ...  Metriplectic "lgebra for Dissipative Fluids in Lagrangian Formulation, submitted. [ ] Morrison, P.J. .  ... 
doi:10.5772/59319 fatcat:yoj6adous5hmtoosqemfpq6e4m

Sub-Fluid Models in Dissipative Magneto-Hydrodynamics [chapter]

Massimo Materassi, Giuseppe Consolini, Emanuele Tassi
2012 Topics in Magnetohydrodynamics  
Morrison of the Institute for Fusion Studies of the University of Texas in Austin, for useful discussions and criticism.  ...  Giuseppe Consolini and Massimo Materassi did this work as a part of ISSI Team n. 185 "Dispersive cascade and dissipation in collisionless space plasma turbulence -observations and simulations", lead by  ...  Some dissipative systems possess however an algebraic structure called metriplectic, which still permits to formulate the dynamics in terms of a bracket and of an observable, extending the concept of Hamiltonian  ... 
doi:10.5772/36022 fatcat:fp5khwe25ja6db32oawxqxsct4

Entropy as a Metric Generator of Dissipation in Complete Metriplectic Systems

Massimo Materassi
2016 Entropy  
This lecture is a short review on the role entropy plays in classical dissipative dynamics formulated in terms of Leibniz bracket algebrae (LBA).  ...  can be given to classical dissipative dynamics through the metriplectic bracket algebra (MBA): the conservative component of the dynamics is still generated via a Poisson algebra by the total energy H  ...  Morrison, who has made me know the beautiful metriplectic formalism, and E. Tassi, who accepted to explore this beautiful subject with me some years ago.  ... 
doi:10.3390/e18080304 fatcat:exbaesh76bh5fppdqxztookt2y

Rayleigh-Bénard Convection as a Nambu-metriplectic problem [article]

Alexander Bihlo
2008 arXiv   pre-print
This leads to a Nambu-metriplectic system, which completes the geometrical picture of RBC.  ...  It is further shown that the equations governing dissipative RBC can be written as the superposition of the conservative Nambu bracket with a dissipative symmetric bracket.  ...  Roman Popovych for the numerous essential discussions. The useful hints by the anonymous referee, which lead to a much improved version of this work are gratefully acknowledged.  ... 
arXiv:0803.4458v2 fatcat:yo46nzniw5awzn42ha42ehlpee

From variational to bracket formulations in nonequilibrium thermodynamics of simple systems [article]

François Gay-Balmaz, Hiroaki Yoshimura
2019 arXiv   pre-print
A similar development has been presented for continuum systems in ElGB2019 and applied to multicomponent fluids.  ...  A variational formulation for nonequilibrium thermodynamics was recently proposed in GBYo2017a,GBYo2017b for both discrete and continuum systems.  ...  F.G.B. is partially supported by the ANR project GEOMFLUID, ANR-14-CE23-0002-01; H.Y. is partially supported by JSPS Grant-in-Aid for Scientific Research (16KT0024, 24224004), the MEXT Top Global University  ... 
arXiv:1904.05958v1 fatcat:qrdzipnd4ze35jwkyoxnh4dd6y

Thoughts on brackets and dissipation: Old and new

P J Morrison
2009 Journal of Physics, Conference Series  
Bracket formulations of two kinds of dynamical systems, called incomplete and complete, are reviewed and developed, including double bracket and metriplectic dynamics.  ...  Dissipation based on the Cartan-Killing metric is introduced. Various examples of incomplete and complete dynamics are discussed, including dynamics associated with three-dimensional Lie algebras.  ...  Introduction The idea of extending the Hamiltonian or Lagrangian framework to include dissipation dates to Rayleigh, but active interest in this idea was renewed in 1981.  ... 
doi:10.1088/1742-6596/169/1/012006 fatcat:dhlyctru6bbgbdiyoanzxe4osy

Metriplectic torque for rotation control of a rigid body [article]

Massimo Materassi, Philip J. Morrison
2018 arXiv   pre-print
Metriplectic dynamics couple a Poisson bracket of the Hamiltonian description with a kind of metric bracket, for describing systems with both Hamiltonian and dissipative components.  ...  In this paper an application of metriplectic dynamics is presented that is of interest for the theory of control: a suitably chosen torque, expressed through a metriplectic extension of its "natural" Poisson  ...  The Euler angles, used as Lagrangian variables for the rigid body. V.  ... 
arXiv:1807.01168v1 fatcat:jbvogiheqndkro2vb3fij3nk6m

Metriplectic torque for rotation control of a rigid body

Massimo Materassi, Philip J. Morrison
2018 Cybernetics and Physics  
Metriplectic dynamics couple a Poisson bracket of the Hamiltonian description with a kind of metric bracket, for describing systems with both Hamiltonian and dissipative components.  ...  In this paper an application of metriplectic dynamics is presented that is of interest for the theory of control: a suitably chosen torque, expressed through a metriplectic extension of its "natural" Poisson  ...  The Euler angles, used as Lagrangian variables for the rigid body.  ... 
doi:10.35470/2226-4116-2018-7-2-78-86 fatcat:vq5nh4ykxzgspecs4vyn4rdiaq

Canonical description of incompressible fluid: Dirac brackets approach

Sonnet Nguyen, Łukasz A. Turski
1999 Physica A: Statistical Mechanics and its Applications  
The resulting equations are then known nonlinear, and non-local in space, equations for incompressible fluid velocity.  ...  This description uses the dynamical constraints, in our case reflecting "incompressibility" assumption, and leads to replacement of usual hydrodynamical Poisson brackets for density and velocity fields  ...  Acknowledgements We would like to thank Cyril Malyshev for contributing discussions during the earlier stage of this work.  ... 
doi:10.1016/s0378-4371(99)00194-6 fatcat:mkllhm4hfvg7bixsauziwko3kq

Recursive properties of Dirac and metriplectic Dirac brackets with applications

Sonnet Hung Q. Nguyen, Łukasz A. Turski
2009 Physica A: Statistical Mechanics and its Applications  
Particular attention is paid to the feasibility of implementation code for Dirac brackets in Computer Algebra System and analytical techniques for inversion of triangular matrices.  ...  We then study the applicability of that formulation in analysis of some interesting physical models.  ...  The main point in metriplectic formulation [13] is that a mixed bracket obtained by adding a symmetric bracket to the Poisson bracket can successfully be used for the description of dissipative systems  ... 
doi:10.1016/j.physa.2008.09.026 fatcat:mibgfxxrzrf55czvheu7qpmu7y

Energetically Consistent Model Reduction for Metriplectic Systems [article]

Anthony Gruber, Max Gunzburger, Lili Ju, Zhu Wang
2022 arXiv   pre-print
Based on proper orthogonal decomposition, a provably convergent metriplectic reduced-order model is formulated which is guaranteed to maintain the algebraic structure necessary for energy conservation  ...  The metriplectic formalism is useful for describing complete dynamical systems which conserve energy and produce entropy.  ...  Moreover, a constrained GENERIC rheological model for polymer solutions was developed in [6] and shown to be effective in predicting steady shear viscosity, while a formulation of dissipative magnetohydrodynamics  ... 
arXiv:2204.08049v1 fatcat:calfkdtxgjcelpvwhj3j77k34i

Structure and structure-preserving algorithms for plasma physics

P. J. Morrison
2017 Physics of Plasmas  
Metriplectic integration, which preserves the structure of conservative systems with both Hamiltonian and dissipative parts, is proposed.  ...  Hamiltonian and action principle (HAP) formulations of plasma physics are reviewed for the purpose of explaining structure preserving numerical algorithms.  ...  of some Lie algebra.  ... 
doi:10.1063/1.4982054 fatcat:hnhphyc46zgl5kyq7zmnk52fim

Thermodynamics of learning physical phenomena [article]

Elias Cueto, Francisco Chinesta
2022 arXiv   pre-print
for the learning process.  ...  We review how thermodynamics provides helpful insights in the learning process.  ...  This type of formulations are also known as metriplectic formulations, since they combine metric and symplectic terms [114, 115] .  ... 
arXiv:2207.12749v1 fatcat:wuthjsru4bfntfkzoazyhxw27e

Dissipation-induced instabilities in finite dimensions

R. Krechetnikov, J. E. Marsden
2007 Reviews of Modern Physics  
The paper contains some new unpublished results and insights, and, finally, open questions are formulated to provide an impetus for future studies.  ...  The theory grounded on the Thomson-Tait-Chetayev and Merkin theorems and on the geometric understanding introduced in this paper leads to the conclusion that ubiquitous dissipation is one of the paramount  ...  This again explains why the Routh method in the case of Abelian symmetries can be understood as having a foot in both the Lagrangian ͑5.3͒ and Hamiltonian ͑5.2͒ formulations.  ... 
doi:10.1103/revmodphys.79.519 fatcat:dz6bjnoncbfenha32qvsobrtg4
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