Topic
Poisson bracket
About: Poisson bracket is a research topic. Over the lifetime, 3990 publications have been published within this topic receiving 85324 citations.
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Papers
Book•
1 Jan 1994
TL;DR: In this paper, the Schouten-Nijenhuis bracket is used for quantization of Poisson manifolds, and the bracket of 1-forms is used to quantize Poisson manifold structures.
Abstract: 0 Introduction.- 1 The Poisson bivector and the Schouten-Nijenhuis bracket.- 1.1 The Poisson bivector.- 1.2 The Schouten-Nijenhuis bracket.- 1.3 Coordinate expressions.- 1.4 The Koszul formula and applications.- 1.5 Miscellanea.- 2 The symplectic foliation of a Poisson manifold.- 2.1 General distributions and foliations.- 2.2 Involutivity and integrability.- 2.3 The case of Poisson manifolds.- 3 Examples of Poisson manifolds.- 3.1 Structures on ?n. Lie-Poisson structures.- 3.2 Dirac brackets.- 3.3 Further examples.- 4 Poisson calculus.- 4.1 The bracket of 1-forms.- 4.2 The contravariant exterior differentiations.- 4.3 The regular case.- 4.4 Cofoliations.- 4.5 Contravariant derivatives on vector bundles.- 4.6 More brackets.- 5 Poisson cohomology.- 5.1 Definition and general properties.- 5.2 Straightforward and inductive computations.- 5.3 The spectral sequence of Poisson cohomology.- 5.4 Poisson homology.- 6 An introduction to quantization.- 6.1 Prequantization.- 6.2 Quantization.- 6.3 Prequantization representations.- 6.4 Deformation quantization.- 7 Poisson morphisms, coinduced structures, reduction.- 7.1 Properties of Poisson mappings.- 7.2 Reduction of Poisson structures.- 7.3 Group actions and momenta.- 7.4 Group actions and reduction.- 8 Symplectic realizations of Poisson manifolds.- 8.1 Local symplectic realizations.- 8.2 Dual pairs of Poisson manifolds.- 8.3 Isotropic realizations.- 8.4 Isotropic realizations and nets.- 9 Realizations of Poisson manifolds by symplectic groupoids.- 9.1 Realizations of Lie-Poisson structures.- 9.2 The Lie groupoid and symplectic structures of T*G.- 9.3 General symplectic groupoids.- 9.4 Lie algebroids and the integrability of Poisson manifolds.- 9.5 Further integrability results.- 10 Poisson-Lie groups.- 10.1 Poisson-Lie and biinvariant structures on Lie groups.- 10.2 Characteristic properties of Poisson-Lie groups.- 10.3 The Lie algebra of a Poisson-Lie group.- 10.4 The Yang-Baxter equations.- 10.5 Manin triples.- 10.6 Actions and dressing transformations.- References.
1,129 citations
TL;DR: In this paper, the concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter, and the formalism generates nonconservative as well as conservative transformations.
Abstract: The concept of a Lie series is enlarged to encompass the cases where the generating function itself depends explicity on the small parameter. Lie transforms define naturally a class of canonical mappings in the form of power series in the small parameter. The formalism generates nonconservative as well as conservative transformations. Perturbation theories based on it offer three substantial advantages: they yield the transformation of state variables in an explicit form; in a function of the original variables, substitution of the new variables consists simply of an iterative procedure involving only explicit chains of Poisson brackets; the inverse transformation can be built the same way.
1,003 citations
TL;DR: In this paper, a general definition of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang-Mills theory and general relativity.
Abstract: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space‐time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the ‘‘field‐independent’’ local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are ‘‘field dependent.’’ As by‐products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.
991 citations
Book•
26 May 1994
TL;DR: In this article, the authors introduce Symplectic geometry in optics and the dynamical theory of liquid crystals for weakly ionized plasma dynamics with respect to the dissipation bracket.
Abstract: PART 1: THEORY Introduction 1. Symplectic geometry in optics 2. Hamiltonian mechanics of discrete particle systems 3. Equilibrium thermodynamics 4. Poisson brackets in continuous media 5. Non-equilibrium thermodynamics 6. The dissipation bracket PART 2: APPLICATIONS 7. Incompressible viscoelastic flows 8. Transport phenomena in viscoelastic fluids 9. Non-standard transport phenomena 10. The dynamical theory of liquid crystals 11. Multi-fluid transport/reactions models with application in the modelling of weakly ionized plasma dynamics Epilogue Bibliography Appendices
903 citations
TL;DR: In this article, the Lie geometric structure behind the Hamiltonian structure of the Korteweg deVries type equations was studied and the authors showed that it is the same as the Lie geometry behind the Lie structure of a Hamiltonian lattice.
Abstract: We study the Lie geometric structure behind the Hamiltonian structure of the Korteweg deVries type equations.
888 citations
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| Year | Papers |
|---|---|
| 2025 | 14 |
| 2024 | 27 |
| 2023 | 80 |
| 2022 | 306 |
| 2021 | 119 |
| 2020 | 127 |