121 results on '"JACOBI identity"'
Search Results
2. Fundamental concepts
- Author
-
Cushman, Richard H., Bates, Larry M., Cushman, Richard H., and Bates, Larry M.
- Published
- 2015
- Full Text
- View/download PDF
3. Poisson Structures in Dimensions Two and Three
- Author
-
Laurent-Gengoux, Camille, Pichereau, Anne, Vanhaecke, Pol, Laurent-Gengoux, Camille, Pichereau, Anne, and Vanhaecke, Pol
- Published
- 2013
- Full Text
- View/download PDF
4. Poisson Structures: Basic Constructions
- Author
-
Laurent-Gengoux, Camille, Pichereau, Anne, Vanhaecke, Pol, Laurent-Gengoux, Camille, Pichereau, Anne, and Vanhaecke, Pol
- Published
- 2013
- Full Text
- View/download PDF
5. Poisson Structures: Basic Definitions
- Author
-
Laurent-Gengoux, Camille, Pichereau, Anne, Vanhaecke, Pol, Laurent-Gengoux, Camille, Pichereau, Anne, and Vanhaecke, Pol
- Published
- 2013
- Full Text
- View/download PDF
6. Group Extensions and the Stages Hypothesis
- Author
-
Morel, J.-M., editor, Takens, F., editor, Teissier, B., editor, Marsden, Jerrold E., Misiolek, Gerard, Ortega, Juan-Pablo, Perlmutter, Matthew, and Ratiu, Tudor S.
- Published
- 2007
- Full Text
- View/download PDF
7. Non-Canonical Hamiltonian Systems
- Author
-
Hairer, Ernst, Wanner, Gerhard, Lubich, Christian, Bank, R., editor, Graham, R.L., editor, Stoer, J., editor, Varga, R., editor, Yserentant, H., editor, Hairer, Ernst, Wanner, Gerhard, and Lubich, Christian
- Published
- 2006
- Full Text
- View/download PDF
8. Lie Algebroids
- Author
-
Dufour, Jean-Paul, Zung, Nguyen Tien, Bass, H., editor, Oesterlé, J., editor, and Weinstein, A., editor
- Published
- 2005
- Full Text
- View/download PDF
9. Symplectic Manifolds
- Author
-
Audin, Michèle, Bass, H., editor, Oesterlé, J., editor, Weinstein, A., editor, and Audin, Michèle
- Published
- 2004
- Full Text
- View/download PDF
10. Poisson Manifolds
- Author
-
Adler, Mark, van Moerbeke, Pierre, Vanhaecke, Pol, Remmert, R., editor, Adler, Mark, van Moerbeke, Pierre, and Vanhaecke, Pol
- Published
- 2004
- Full Text
- View/download PDF
11. Classical Equations
- Author
-
Jackiw, Roman and Jackiw, Roman
- Published
- 2002
- Full Text
- View/download PDF
12. Further Topics in Structure Preservation
- Author
-
Hairer, Ernst, Wanner, Gerhard, Lubich, Christian, Hairer, Ernst, Wanner, Gerhard, and Lubich, Christian
- Published
- 2002
- Full Text
- View/download PDF
13. Dual Algebras with Non-Linear Poisson Brackets
- Author
-
Korovnichenko, A., Zhedanov, A., Pakuliak, S., editor, and von Gehlen, G., editor
- Published
- 2001
- Full Text
- View/download PDF
14. Hamiltonian Formulations
- Author
-
Jou, David, Casas-Vázquez, José, Lebon, Georgy, Jou, David, Casas-Vázquez, José, and Lebon, Georgy
- Published
- 2001
- Full Text
- View/download PDF
15. Normal Forms and Symmetries for Hamiltonian Systems
- Author
-
Beig, R., editor, Ehlers, J., editor, Frisch, U., editor, Hepp, K., editor, Jaffe, R. L., editor, Kippenhahn, R., editor, Ojima, I., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, Beiglböck, W., editor, Cicogna, Giampaolo, and Gaeta, Giuseppe
- Published
- 1999
- Full Text
- View/download PDF
16. Poisson brackets for densities of functionals
- Author
-
Dickey, Leonid A., Araki, H., editor, Beig, R., editor, Ehlers, J., editor, Frisch, U., editor, Hepp, K., editor, Jaffe, R. L., editor, Kippenhahn, R., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, Beiglböck, W., editor, Lehr, Sabine, editor, Aratyn, Henrik, editor, Imbo, Tom D., editor, Keung, Wai-Yee, editor, and Sukhatme, Uday, editor
- Published
- 1998
- Full Text
- View/download PDF
17. W-algebras and their representations
- Author
-
Watts, Gerard M. T., Araki, H., editor, Beig, R., editor, Ehlers, J., editor, Frisch, U., editor, Hepp, K., editor, Jaffe, R. L., editor, Kippenhahn, R., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, Beiglböck, W., editor, Horváth, Zalán, editor, and Palla, László, editor
- Published
- 1997
- Full Text
- View/download PDF
18. Construction of Hamiltonian Structures for Dynamical Systems from Scratch
- Author
-
Hojman, Sergio A., Tirapegui, Enrique, editor, Martinez, Servet, editor, and Zeller, Walter, editor
- Published
- 1996
- Full Text
- View/download PDF
19. Outline of a theory of the KdV equation
- Author
-
Lax, Peter D., Boillat, Guy, Dafermos, Constantin M., Lax, Peter D., Liu, Tai-Ping, and Ruggeri, Tommaso, editor
- Published
- 1996
- Full Text
- View/download PDF
20. Hamiltonian Structure of Multicomponent KdV Equations
- Author
-
Oğuz, Ömer and Gruber, Bruno, editor
- Published
- 1995
- Full Text
- View/download PDF
21. Nematic Liquid Crystals
- Author
-
Kats, E. I., Lebedev, V. V., Kats, E. I., and Lebedev, V. V.
- Published
- 1994
- Full Text
- View/download PDF
22. Elements of Differential Equations
- Author
-
Iwasaki, Katsunori, Kimura, Hironobu, Shimomura, Shun, Yoshida, Masaaki, Diederich, Klas, editor, Iwasaki, Katsunori, Kimura, Hironobu, Shimomura, Shun, and Yoshida, Masaaki
- Published
- 1991
- Full Text
- View/download PDF
23. Introduction to quantum groups
- Author
-
Takhtajan, L. A., Araki, H., editor, Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Ruelle, D., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, Beiglböck, W., editor, Doebner, H. -D., editor, and Hennig, J. -D., editor
- Published
- 1990
- Full Text
- View/download PDF
24. Mathematical guide to quantum groups
- Author
-
Doebner, H. D., Hennig, J. D., Lücke, W., Araki, H., editor, Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Ruelle, D., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, Beiglböck, W., editor, Doebner, H. -D., editor, and Hennig, J. -D., editor
- Published
- 1990
- Full Text
- View/download PDF
25. Nambu dynamics and its noncanonical Hamiltonian representation in many degrees of freedom systems
- Author
-
Atsushi Horikoshi
- Subjects
Jacobi identity ,High Energy Physics - Theory ,General Physics and Astronomy ,Semiclassical physics ,FOS: Physical sciences ,Physics - Classical Physics ,01 natural sciences ,Poisson bracket ,symbols.namesake ,Identity (mathematics) ,0103 physical sciences ,010306 general physics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematical physics ,Physics ,Hamiltonian mechanics ,010308 nuclear & particles physics ,Degrees of freedom ,High Energy Physics::Phenomenology ,Classical Physics (physics.class-ph) ,Mathematical Physics (math-ph) ,Bracket (mathematics) ,High Energy Physics - Theory (hep-th) ,symbols ,Hamiltonian (control theory) - Abstract
Nambu dynamics is a generalized Hamiltonian dynamics of more than two variables, whose time evolutions are given by the Nambu bracket, a generalization of the canonical Poisson bracket. Nambu dynamics can always be represented in the form of noncanonical Hamiltonian dynamics by defining the noncanonical Poisson bracket by means of the Nambu bracket. For the time evolution to be consistent, the Nambu bracket must satisfy the fundamental identity, while the noncanonical Poisson bracket must satisfy the Jacobi identity. However, in many degrees of freedom systems, it is well known that the fundamental identity does not hold. In the present paper we show that, even if the fundamental identity is violated, the Jacobi identity for the corresponding noncanonical Hamiltonian dynamics could hold. As an example, we evaluate these identities for a semiclassical system of two coupled oscillators., 7 pages, final version
- Published
- 2021
26. Generalization of Hamiltonian Mechanics to a Three Dimensional Phase Space
- Author
-
Naoki Sato
- Subjects
Hamiltonian mechanics ,Jacobi identity ,Physics ,Constant of motion ,General Physics and Astronomy ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Action (physics) ,symbols.namesake ,Poisson bracket ,Phase space ,symbols ,Mathematics::Symplectic Geometry ,Hamiltonian (control theory) ,Mathematical Physics ,Mathematical physics ,Symplectic geometry - Abstract
Classical Hamiltonian mechanics is realized by the action of a Poisson bracket on a Hamiltonian function. The Hamiltonian function is a constant of motion (the energy) of the system. The properties of the Poisson bracket are encapsulated in the symplectic 2-form, a closed second order differential form. Due to closure, the symplectic 2-form is preserved by the Hamiltonian flow, and it assigns an invariant (Liouville) measure on the phase space through the Lie-Darboux theorem. In this paper we propose a generalization of classical Hamiltonian mechanics to a three-dimensional phase space: the classical Poisson bracket is replaced with a generalized Poisson bracket acting on a pair of Hamiltonian functions, while the symplectic 2-form is replaced by a symplectic 3-form. We show that, using the closure of the symplectic 3-form, a result analogous to the classical Lie-Darboux theorem holds: locally, there exist smooth coordinates such that the components of the symplectic 3-form are constants, and the phase space is endowed with a preserved volume element. Furthermore, as in the classical theory, the Jacobi identity for the generalized Poisson bracket mathematically expresses the closure of the associated symplectic form. As a consequence, constant skew-symmetric third order contravariant tensors always define generalized Poisson brackets. This is in contrast with generalizations of Hamiltonian mechanics postulating the fundamental identity as replacement for the Jacobi identity. In particular, we find that the fundamental identity represents a stronger requirement than the closure of the symplectic 3-form., 16 pages
- Published
- 2020
27. Three computational approaches to weakly nonlocal Poisson brackets
- Author
-
Casati, Matteo, Lorenzoni, Paolo, Vitolo, Raffaele, Casati, M, Lorenzoni, P, Vitolo, R, Casati, M., Lorenzoni, P., and Vitolo, R.
- Subjects
Jacobi identity ,Vertex (graph theory) ,Pure mathematics ,Computation ,FOS: Physical sciences ,Poisson distribution ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Poisson bracket ,0103 physical sciences ,0101 mathematics ,solitons and integrable systems ,Mathematical Physics ,Mathematics ,Partial differential equation ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Pseudodifferential operators ,Applied Mathematics ,010102 general mathematics ,Mathematical Physics (math-ph) ,mathematical physic ,partial differential equation ,symbols ,37K05, 35S05 ,Exactly Solvable and Integrable Systems (nlin.SI) - Abstract
We compare three different ways of checking the Jacobi identity for weakly nonlocal Poisson brackets using the theory of distributions, of pseudodifferential operators and of Poisson vertex algebras, respectively. We show that the three approaches lead to similar computations and same results., 47 pages
- Published
- 2020
- Full Text
- View/download PDF
28. The replicator dynamics of zero-sum games arise from a novel poisson algebra
- Author
-
Christopher Griffin
- Subjects
Jacobi identity ,Physics - Physics and Society ,Pure mathematics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,General Mathematics ,Applied Mathematics ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Physics and Society (physics.soc-ph) ,Function (mathematics) ,Conserved quantity ,Poisson bracket ,symbols.namesake ,Bracket (mathematics) ,Mathematics - Symplectic Geometry ,Replicator equation ,FOS: Mathematics ,symbols ,Symplectic Geometry (math.SG) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Mathematics::Symplectic Geometry ,Poisson algebra ,Mathematics ,Symplectic geometry - Abstract
We show that the replicator dynamics for zero-sum games arises as a result of a non-canonical bracket that is a hybrid between a Poisson Bracket and a Nambu Bracket. The resulting non-canonical bracket is parameterized both the by the skew-symmetric payoff matrix and a mediating function. The mediating function is only sometimes a conserved quantity, but plays a critical role in the determination of the dynamics. As a by-product, we show that for the replicator dynamics this function arises in the definition of a natural metric on which phase flow-volume is preserved. Additionally, we show that the non-canonical bracket satisfies all the same identities as the Poisson bracket except for the Jacobi identity (JI), which is satisfied for special cases of the mediating function. In particular, the mediating function that gives rise to the replicator dynamics yields a bracket that satisfies JI. This neatly explains why the mediating function allows us to derive a metric on which phase flow is conserved and suggests a natural geometry for zero-sum games that extends the Symplectic geometry of the Poisson bracket and potentially an alternate approach to quantizing evolutionary games., 13 pages, 2 figures
- Published
- 2021
29. On Hamiltonian continuum mechanics
- Author
-
Ilya Peshkov, Václav Klika, and Michal Pavelka
- Subjects
Jacobi identity ,Inertial frame of reference ,FOS: Physical sciences ,Physics - Classical Physics ,01 natural sciences ,010305 fluids & plasmas ,symbols.namesake ,Poisson bracket ,0103 physical sciences ,010306 general physics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Physics ,Hamiltonian mechanics ,Continuum mechanics ,Fluid Dynamics (physics.flu-dyn) ,Classical Physics (physics.class-ph) ,Statistical and Nonlinear Physics ,Eulerian path ,Physics - Fluid Dynamics ,Mathematical Physics (math-ph) ,Condensed Matter Physics ,Classical mechanics ,Finite strain theory ,symbols ,Hamiltonian (quantum mechanics) - Abstract
Continuum mechanics can be formulated in the Lagrangian frame (addressing motion of individual continuum particles) or in the Eulerian frame (addressing evolution of fields in an inertial frame). There is a canonical Hamiltonian structure in the Lagrangian frame. By transformation to the Eulerian frame we find the Poisson bracket for Eulerian continuum mechanics with deformation gradient (or the related distortion matrix). Both Lagrangian and Eulerian Hamiltonian structures are then discussed from the perspective of space-time variational formulation and by means of semidirect products and Lie algebras. Finally, we discuss the importance of the Jacobi identity in continuum mechanics and approaches to prove hyperbolicity of the evolution equations and their gauge invariance., Comment: Submitted to Physica D
- Published
- 2019
- Full Text
- View/download PDF
30. Plasma in a monopole background does not have a twisted Poisson structure
- Author
-
C. Sardón, Alan J. Weinstein, Manuel Lainz, and UAM. Departamento de Matemáticas
- Subjects
Jacobi identity ,Matemáticas ,Poisson distribution ,Space (mathematics) ,01 natural sciences ,Atomic ,symbols.namesake ,Poisson bracket ,Particle and Plasma Physics ,Poisson manifold ,Mathematics::Quantum Algebra ,0103 physical sciences ,Nuclear ,State space (physics) ,010306 general physics ,Mathematics::Symplectic Geometry ,Mathematical physics ,Physics ,Quantum Physics ,010308 nuclear & particles physics ,Molecular ,Nuclear & Particles Physics ,Bracket (mathematics) ,Phase space ,symbols ,Astronomical and Space Sciences - Abstract
Author(s): Lainz, M; Sardon, C; Weinstein, A | Abstract: For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding bracket operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson bracket; however, it is twisted Poisson in the sense that the Jacobiator comes from a closed 3-form. The space D of densities on phase space is the state space of a plasma. The twisted Poisson bracket on phase-space functions gives rise to a bracket on functions on D. In the absence of monopoles, this is again a Poisson bracket. It has recently been shown by Heninger and Morrison that this bracket is not Poisson when monopoles are present. In this note, we give an example where it is not even twisted Poisson.
- Published
- 2019
31. The Lie algebra of classical mechanics
- Author
-
Ander Murua and Robert I. McLachlan
- Subjects
Jacobi identity ,Physics ,17B01, 65P10, 70H05 ,Direct sum ,Computational Mechanics ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Numerical Analysis (math.NA) ,Linear subspace ,Computational Mathematics ,symbols.namesake ,Poisson bracket ,Classical mechanics ,Euclidean geometry ,Lie algebra ,symbols ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Abelian group ,Commutative property ,Mathematical Physics - Abstract
Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the `Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra $\mathcal X$, spanned by `modified' potential energies isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with $\mathcal X$. We calculate the dimensions $c_n$ of its homogeneous subspaces and determine the value of its entropy $\lim_{n\to\infty} c_n^{1/n}$. It is $1.8249\dots$, a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics., Comment: 17 pages, submitted to Journal of Computational Dynamics
- Published
- 2019
- Full Text
- View/download PDF
32. Dissipative effects in magnetohydrodynamical models with intrinsic magnetization
- Author
-
Manasvi Lingam
- Subjects
Physics ,Jacobi identity ,Hamiltonian mechanics ,Numerical Analysis ,Angular momentum ,Entropy production ,Applied Mathematics ,Laws of thermodynamics ,symbols.namesake ,Poisson bracket ,Classical mechanics ,Modeling and Simulation ,Dissipative system ,symbols ,Spin-½ - Abstract
A unifying non-canonical Poisson bracket has been shown to describe magnetohydrodynamic models of classical and quantum–mechanical fluids with intrinsic magnetization (or spin), and the Jacobi identity for this bracket is also proven. Their corresponding Hamiltonians are presented, and some interesting features, such as the potential absence of angular momentum conservation, are pointed out. To maintain consistency with the first and second laws of thermodynamics, a metriplectic approach to these models is highlighted which involves entropy production via a hitherto unstudied term involving the magnetization. A few promising avenues and outstanding issues for future work in this area are also discussed.
- Published
- 2015
33. On deformations of classical mechanics due to Planck-scale physics
- Author
-
Abhijit Sen, Olga Chashchina, Z.K. Silagadze, and École polytechnique (X)
- Subjects
High Energy Physics - Theory ,Jacobi identity ,Poisson bracket ,Uncertainty principle ,Koopman–von Neumann theory ,FOS: Physical sciences ,Physics - Classical Physics ,mechanics: classical ,String theory ,01 natural sciences ,uncertainty relations: Heisenberg ,symbols.namesake ,0103 physical sciences ,Quantum system ,commutation relations ,classical-quantum dynamics ,010306 general physics ,Quantum ,Mathematical Physics ,Physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,010308 nuclear & particles physics ,quantum mechanics ,Hilbert space ,scale: Planck ,Classical Physics (physics.class-ph) ,Astronomy and Astrophysics ,generalized commutation relations ,[PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph] ,Classical mechanics ,hidden variable ,High Energy Physics - Theory (hep-th) ,quantum gravity ,Space and Planetary Science ,Hidden variable theory ,symbols ,Quantum gravity - Abstract
Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman-Von Neumann's Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations., Comment: 39 pages, ws-ijmpd, version to be published in Int. J. Mod. Phys. D
- Published
- 2020
34. Numerical simulations of one laser-plasma model based on Poisson structure
- Author
-
Yajuan Sun, Nicolas Crouseilles, and Yingzhe Li
- Subjects
Jacobi identity ,Physics ,Numerical Analysis ,Finite volume method ,Physics and Astronomy (miscellaneous) ,Discretization ,Applied Mathematics ,Discrete Poisson equation ,Mathematical analysis ,010103 numerical & computational mathematics ,Poisson distribution ,01 natural sciences ,Computer Science Applications ,010101 applied mathematics ,Computational Mathematics ,symbols.namesake ,Poisson bracket ,Modeling and Simulation ,Poisson manifold ,symbols ,0101 mathematics ,Poisson's equation - Abstract
In this paper, a bracket structure is proposed for the laser-plasma interaction model introduced in [19], and it is proved by direct calculations that the bracket is Poisson which satisfies the Jacobi identity. Then splitting methods in time are proposed based on the Poisson structure. For the quasi-relativistic case, the Hamiltonian splitting leads to three subsystems which can be solved exactly. The conservative splitting is proposed for the fully relativistic case, and three one-dimensional conservative subsystems are obtained. Combined with the splittings in time, in phase space discretization we use the Fourier spectral and finite volume methods. It is proved that the discrete charge and discrete Poisson equation are conserved by our numerical schemes. Numerically, some numerical experiments are conducted to verify good conservations for the charge, energy and Poisson equation.
- Published
- 2020
35. Hamiltonian nature of monopole dynamics
- Author
-
Philip J. Morrison and J. M. Heninger
- Subjects
Jacobi identity ,Physics ,High Energy Physics::Lattice ,Hamiltonian field theory ,Magnetic monopole ,Canonical coordinates ,General Physics and Astronomy ,01 natural sciences ,010305 fluids & plasmas ,Quantization (physics) ,symbols.namesake ,Poisson bracket ,0103 physical sciences ,symbols ,Classical electromagnetism ,Quantum field theory ,010306 general physics ,Mathematical physics - Abstract
Classical electromagnetism with magnetic monopoles is not a Hamiltonian field theory because the Jacobi identity for the Poisson bracket fails. The Jacobi identity is recovered only if all of the species have the same ratio of electric to magnetic charge or if an electron and a monopole can never collide. Without the Jacobi identity, there are no local canonical coordinates or Lagrangian action principle. To build a quantum field of magnetic monopoles, we either must explain why the positions of electrons and monopoles can never coincide or we must resort to new quantization techniques.
- Published
- 2020
36. Automated symbolic calculations in nonequilibrium thermodynamics
- Author
-
Kröger, Martin and Hütter, Markus
- Subjects
- *
JACOBI identity , *AUTOMATION , *NONEQUILIBRIUM thermodynamics , *POISSON brackets , *NONLINEAR differential equations , *SYSTEM integration - Abstract
Abstract: We cast the Jacobi identity for continuous fields into a local form which eliminates the need to perform any partial integration to the expense of performing variational derivatives. This allows us to test the Jacobi identity definitely and efficiently and to provide equations between different components defining a potential Poisson bracket. We provide a simple MathematicaTM notebook which allows to perform this task conveniently, and which offers some additional functionalities of use within the framework of nonequilibrium thermodynamics: reversible equations of change for fields, and the conservation of entropy during the reversible dynamics. Program summary: Program title: Poissonbracket.nb Catalogue identifier: AEGW_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEGW_v1_0.html Program obtainable from: CPC Program Library, Queen''s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 227 952 No. of bytes in distributed program, including test data, etc.: 268 918 Distribution format: tar.gz Programming language: MathematicaTM 7.0 Computer: Any computer running MathematicaTM 6.0 and later versions Operating system: Linux, MacOS, Windows RAM: 100 Mb Classification: 4.2, 5, 23 Nature of problem: Testing the Jacobi identity can be a very complex task depending on the structure of the Poisson bracket. The MathematicaTM notebook provided here solves this problem using a novel symbolic approach based on inherent properties of the variational derivative, highly suitable for the present tasks. As a by product, calculations performed with the Poisson bracket assume a compact form. Solution method: The problem is first cast into a form which eliminates the need to perform partial integration for arbitrary functionals at the expense of performing variational derivatives. The corresponding equations are conveniently obtained using the symbolic programming environment MathematicaTM. Running time: For the test cases and most typical cases in the literature, the running time is of the order of seconds or minutes, respectively. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
37. Poisson brackets after Jacobi and Plucker
- Author
-
Damianou, Pantelis A. and Damianou, Pantelis A. [0000-0003-3399-9837]
- Subjects
Jacobi identity ,Mathematics - Differential Geometry ,Pure mathematics ,Rank (linear algebra) ,FOS: Physical sciences ,02 engineering and technology ,01 natural sciences ,symbols.namesake ,Poisson bracket ,Mathematics (miscellaneous) ,0202 electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,Projective space ,0101 mathematics ,Plucker ,Mathematics::Symplectic Geometry ,37J35, 53D17 ,Mathematical Physics ,Mathematics ,Mechanical Engineering ,Applied Mathematics ,010102 general mathematics ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Jacobi elliptic functions ,Differential Geometry (math.DG) ,Modeling and Simulation ,symbols ,020201 artificial intelligence & image processing ,Realization (systems) ,Symplectic geometry - Abstract
We construct a symplectic realization and a bi-hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems., Comment: 19 pages
- Published
- 2018
- Full Text
- View/download PDF
38. Hamiltonian structure of the guiding center plasma model
- Author
-
Joshua W. Burby and Wrick Sengupta
- Subjects
Physics ,Jacobi identity ,Guiding center ,Vlasov equation ,FOS: Physical sciences ,Condensed Matter Physics ,01 natural sciences ,Physics - Plasma Physics ,010305 fluids & plasmas ,Hamiltonian system ,Plasma Physics (physics.plasm-ph) ,Poisson bracket ,symbols.namesake ,Classical mechanics ,Maxwell's equations ,Physics::Plasma Physics ,0103 physical sciences ,Physics::Space Physics ,symbols ,Poisson's equation ,010306 general physics ,Hamiltonian (quantum mechanics) - Abstract
The guiding center plasma model (also known as kinetic MHD) is a rigorous sub-cyclotron-frequency closure of the Vlasov-Maxwell system. While the model has been known for decades, and it plays a fundamental role in describing the physics of strongly-magnetized collisionless plasmas, its Hamiltonian structure has never been found. We provide explicit expressions for the model's Poisson bracket and Hamiltonian, and thereby prove that the model is an infinite-dimensional Hamiltonian system. The bracket is derived in a manner that ensures it satisfies the Jacobi identity. We also report on several previously-unknown circulation theorems satisfied by the guiding center plasma model. Without knowledge of the Hamiltonian structure, these circulation theorems would be difficult to guess., 5 pages with references, v. 2.0
- Published
- 2017
39. Lifting a weak Poisson bracket to the algebra of forms
- Author
-
Simon L. Lyakhovich, Alexey A. Sharapov, and Matthew T. Peddie
- Subjects
Hamiltonian mechanics ,Jacobi identity ,Mathematics - Differential Geometry ,Pure mathematics ,Пуассона скобки ,General Physics and Astronomy ,FOS: Physical sciences ,53D17, 53Z05, 58A50 ,Mathematical Physics (math-ph) ,Poisson distribution ,Submanifold ,BRST quantization ,symbols.namesake ,Poisson bracket ,Differential Geometry (math.DG) ,Poisson manifold ,symbols ,FOS: Mathematics ,алгебра форм ,Geometry and Topology ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,First class constraint ,Mathematical Physics ,Mathematics - Abstract
We detail the construction of a weak Poisson bracket over a submanifold Σ of a smooth manifold M with respect to a local foliation of this submanifold. Such a bracket satisfies a weak type Jacobi identity but may be viewed as a usual Poisson bracket on the space of leaves of the foliation. We then lift this weak Poisson bracket to a weak odd Poisson bracket on the odd tangent bundle ΠTM, interpreted as a weak Koszul bracket on differential forms on M. This lift is achieved by encoding the weak Poisson structure into a homotopy Poisson structure on an extended manifold, and lifting the Hamiltonian function that generates this structure. Such a construction has direct physical interpretation. For a generic gauge system, the submanifold Σ may be viewed as a stationary surface or a constraint surface, with the foliation given by the foliation of the gauge orbits. Through this interpretation, the lift of the weak Poisson structure is simply a lift of the action generating the corresponding BRST operator of the system.
- Published
- 2017
40. GEMPIC: geometric electromagnetic particle-in-cell methods
- Author
-
Eric Sonnendrücker, Philip J. Morrison, Michael Kraus, and Katharina Kormann
- Subjects
Jacobi identity ,Discretization ,FOS: Physical sciences ,010103 numerical & computational mathematics ,01 natural sciences ,010305 fluids & plasmas ,Hamiltonian system ,symbols.namesake ,Poisson bracket ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Physics ,Charge conservation ,Mathematical analysis ,Numerical Analysis (math.NA) ,Computational Physics (physics.comp-ph) ,Condensed Matter Physics ,Finite element method ,Physics - Plasma Physics ,Plasma Physics (physics.plasm-ph) ,Finite element exterior calculus ,symbols ,Hamiltonian (quantum mechanics) ,Physics - Computational Physics - Abstract
We present a novel framework for Finite Element Particle-in-Cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive a semi-discrete Poisson bracket, which retains the defining properties of a bracket, anti-symmetry and the Jacobi identity, as well as conservation of its Casimir invariants, implying that the semi-discrete system is still a Hamiltonian system. In order to obtain a fully discrete Poisson integrator, the semi-discrete bracket is used in conjunction with Hamiltonian splitting methods for integration in time. Techniques from Finite Element Exterior Calculus ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge invariant, feature exact charge conservation and show excellent long-time energy and momentum behaviour. Due to the generality of our framework, these conservation properties are guaranteed independently of a particular choice of the Finite Element basis, as long as the corresponding Finite Element spaces satisfy certain compatibility conditions., 57 Pages
- Published
- 2017
- Full Text
- View/download PDF
41. The field equations in a three-dimensional commutative space
- Author
-
Su Long Nyeo and Mu Yi Chen
- Subjects
Physics ,Jacobi identity ,Nuclear and High Energy Physics ,010308 nuclear & particles physics ,Astronomy and Astrophysics ,Space (mathematics) ,01 natural sciences ,Atomic and Molecular Physics, and Optics ,Momentum ,symbols.namesake ,Poisson bracket ,Maxwell's equations ,Position (vector) ,0103 physical sciences ,symbols ,Commutation ,010306 general physics ,Commutative property ,Mathematical physics - Abstract
The field equations in a three-dimensional commutative space based on a set of commutation relations are derived. In this space, the commutation relation of the position and kinematic momentum of a particle is generalized to include a metric tensor field in addition to a vector field. The introduction of a metric tensor is a generalization of the commutation relation for Feynman’s proof of the Maxwell equations. In this paper, as the equations of motion and the field equations are classical, the Poisson bracket and not the commutation relation is used in the calculations. As the commutative space is defined by the Poisson bracket, the equations of motion for the particle and the field equations for the metric tensor and vector are derived from the Poisson bracket in Hamiltonian mechanics. The Helmholtz conditions, which express the existence of a Lagrangian for a particle in the space, are also derived from the Poisson bracket. Then the field equations are calculated explicitly by two approaches. One is to calculate the Helmholtz conditions using the equations of motion. The other is to calculate the Jacobi identity for the kinematic momentum or velocity of the particle. In addition to the homogeneous Maxwell equations, the generalized field equations are obtained to define the generalized electric and magnetic fields of the tensor field. Just like the usual electric and magnetic fields, the generalized fields are invariant under a local gauge transformation and should play significant roles in physics. Finally, the homogeneous Maxwell equations of the vector field are seen to exhibit similarities with the generalized field equations for the tensor field. This similarity provides a useful theoretical framework for constructing gravitoelectromagnetism, which is based on analogies between the equations for electromagnetism and relativistic gravitation. It remains to establish the usefulness of the theoretical framework with applications of the field equations.
- Published
- 2019
42. Nonassociativity, Malcev algebras and string theory
- Author
-
Djordje Minic and Murat Gunaydin
- Subjects
High Energy Physics - Theory ,Jacobi identity ,Field (physics) ,Magnetic monopole ,FOS: Physical sciences ,General Physics and Astronomy ,Duality (optimization) ,String theory ,01 natural sciences ,Mathematics::Group Theory ,Poisson bracket ,Theoretical physics ,symbols.namesake ,0103 physical sciences ,D-brane ,0101 mathematics ,Mathematical Physics ,Physics ,Quantum Physics ,010308 nuclear & particles physics ,Mathematics::Rings and Algebras ,010102 general mathematics ,Mathematical Physics (math-ph) ,String field theory ,16. Peace & justice ,High Energy Physics - Theory (hep-th) ,symbols ,Quantum Physics (quant-ph) - Abstract
Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality., 33 pages; Latex file; References added, minor changes, typos corrected
- Published
- 2013
43. Classical r-matrices, lax equations, poisson lie groups and dressing transformations
- Author
-
Semenov-Tian-Shansky, M. A., Araki, H., editor, Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittarts, J., editor, Beiglböck, W., editor, de Vega, H. J., editor, and Sánchez, N., editor
- Published
- 1987
- Full Text
- View/download PDF
44. Concatenation of Lie algebraic maps
- Author
-
Healy, Liam M., Dragt, Alex J., Araki, H., editor, Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Ruelle, D., editor, Weidenmüller, H. A., editor, Wess, J., editor, Zittartz, J., editor, Beiglböck, W., editor, and Wolf, Kurt Bernardo, editor
- Published
- 1989
- Full Text
- View/download PDF
45. Yang-Baxter charge algebras in integrable classical and quantum field theorie
- Author
-
Eichenherr, H., de Vega, H. J., Maillet, J. M., Araki, H., editor, Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Weindenmüller, H. A., editor, Zittartz, J., editor, Beiglböck, W., editor, and Sanchez, N., editor
- Published
- 1985
- Full Text
- View/download PDF
46. Mastersymmetries for completely integrable systems in statistical mechanics
- Author
-
Fuchssteiner, Benno, Araki, H., editor, Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Weidenmüller, H. A., editor, Zittartz, J., editor, and Garrido, Luis, editor
- Published
- 1985
- Full Text
- View/download PDF
47. Hamiltonian equations of hydrodynamics of quantum liquid in the presence of solitons
- Author
-
Khalatnikov, I. M., Lebedev, V. V., Ehlers, J., editor, Hepp, K., editor, Kippenhahn, R., editor, Weidenmüller, H. A., editor, Zittartz, J., editor, Beiglböck, W., editor, Pękalski, Andrzej, editor, and Przystawa, Jerzy A., editor
- Published
- 1980
- Full Text
- View/download PDF
48. Hamiltonian Methods for Evolution Equations
- Author
-
Olver, Peter J., Halmos, P. R., editor, Gehring, F. W., editor, Moore, C. C., editor, and Olver, Peter J.
- Published
- 1986
- Full Text
- View/download PDF
49. Finite-Dimensional Hamiltonian Systems
- Author
-
Olver, Peter J., Halmos, P. R., editor, Gehring, F. W., editor, Moore, C. C., editor, and Olver, Peter J.
- Published
- 1986
- Full Text
- View/download PDF
50. Symplectic Group, Quantum Mechanics and Anosov’s Systems
- Author
-
Avez, André, Blaquiére, A., editor, Fer, F., editor, and Marzollo, A., editor
- Published
- 1980
- Full Text
- View/download PDF
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.