Relevant bibliographies by topics / Symplectic and Poisson geometry

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Symplectic and Poisson geometry'

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Published: 11 March 2023

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Journal articles on the topic "Symplectic and Poisson geometry"

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BOJOWALD, MARTIN, and THOMAS STROBL. "POISSON GEOMETRY IN CONSTRAINED SYSTEMS." Reviews in Mathematical Physics 15, no. 07 (September 2003): 663–703. http://dx.doi.org/10.1142/s0129055x0300176x.

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Associated to a constrained system with closed constraint algebra there are two Poisson manifolds P and Q forming a symplectic dual pair with respect to the original, unconstrained phase space: P is the image of the constraint map (equipped with the algebra of constraints) and Q the Poisson quotient with respect to the orbits generated by the constraints (the orbit space is assumed to be a manifold). We provide sufficient conditions so that the reduced phase space of the constrained system may be identified with a symplectic leaf of Q. By these methods, a second class constrained system with closed algebra is reformulated as an abelian first class system in an extended phase space. While any Poisson manifold (P,Π) has a symplectic realization (Karasev, Weinstein 87), it does not always permit a leafwise symplectic embedding into a symplectic manifold (M,ω). For regular P, it is seen that such an embedding exists, iff the characteristic form-class of Π, a certain element of the third relative cohomology of P, vanishes. A tubular neighborhood of the constraint surface of a general second class constrained system equipped with the Dirac bracket provides a physical example for such an embedding into the original symplectic manifold. In contrast, a leafwise symplectic embedding of e.g. (the maximal regular part of) a Poisson Lie manifold associated to a compact, semisimple Lie algebra does not exist.
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Contreras, Ivan, and Nicolás Martínez Alba. "Poly-symplectic geometry and the AKSZ formalism." Reviews in Mathematical Physics 33, no. 09 (May 31, 2021): 2150030. http://dx.doi.org/10.1142/s0129055x21500306.

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In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.
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Cahen, Michel, and LORENZ J. SCHWACHH�FER. "Special Symplectic Connections and Poisson Geometry." Letters in Mathematical Physics 69, no. 1-3 (July 2004): 115–37. http://dx.doi.org/10.1007/s11005-004-0474-4.

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Crooks, Peter, and Markus Röser. "The $\log$ symplectic geometry of Poisson slices." Journal of Symplectic Geometry 20, no. 1 (2022): 135–90. http://dx.doi.org/10.4310/jsg.2022.v20.n1.a4.

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Guillemin, Victor, Eva Miranda, and Ana Rita Pires. "Symplectic and Poisson geometry on b-manifolds." Advances in Mathematics 264 (October 2014): 864–96. http://dx.doi.org/10.1016/j.aim.2014.07.032.

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Ortega, Juan-Pablo, and Judor S. Ratiu. "Symmetry Reduction in Symplectic and Poisson Geometry." Letters in Mathematical Physics 69, no. 1-3 (July 2004): 11–60. http://dx.doi.org/10.1007/s11005-004-0898-x.

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Ivancevic, V., and C. E. M. Pearce. "Poisson manifolds in generalised Hamiltonian biomechanics." Bulletin of the Australian Mathematical Society 63, no. 3 (June 2001): 515–26. http://dx.doi.org/10.1017/s0004972700019584.

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In this paper the generalised Hamiltonian approach to the modelling of dynamical systems is developed no via the standard formalism of symplectic geometry but rather via Poisson manifolds and evolution equations. This alternative approach has the merit of being available in a wider context than the former. Application is made to three biomechanical models, one in which the symplectic–geometry approach also applies (the motion of a body segment) and two in which it does not (Schwan's model of blood and lymph circulation and Davydov's molecular model of muscle contraction).
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ASADI, E., and J. A. SANDERS. "INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY." Glasgow Mathematical Journal 51, A (February 2009): 5–23. http://dx.doi.org/10.1017/s0017089508004746.

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AbstractQuaternionic vector mKDV equations are derived from the Cartan structure equation in the symmetric space=Sp(n+1)/Sp(1) ×Sp(n). The derivation of the soliton hierarchy utilizes a moving parallel frame and a Cartan connection 1-form ω related to the Cartan geometry onmodelled on$(\mk{sp}_{n+1}, \mk{sp}_{1}\,{\times}\, \mk{sp}_{n})$. The integrability structure is shown to be geometrically encoded by a Poisson–Nijenhuis structure and a symplectic operator.
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LAVROV, P. M., and O. V. RADCHENKO. "SYMPLECTIC GEOMETRIES ON SUPERMANIFOLDS." International Journal of Modern Physics A 23, no. 09 (April 10, 2008): 1337–50. http://dx.doi.org/10.1142/s0217751x08039426.

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Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with a nondegenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of two different types of odd geometries on supermanifolds.
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Frejlich, Pedro, and Ioan Mărcuț. "The Homology Class of a Poisson Transversal." International Mathematics Research Notices 2020, no. 10 (May 23, 2018): 2952–76. http://dx.doi.org/10.1093/imrn/rny105.

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Abstract This note is devoted to the study of the homology class of a compact Poisson transversal in a Poisson manifold. For specific classes of Poisson structures, such as unimodular Poisson structures and Poisson manifolds with closed leaves, we prove that all their compact Poisson transversals represent nontrivial homology classes, generalizing the symplectic case. We discuss several examples in which this property does not hold, as well as a weaker version of this property, which holds for log-symplectic structures. Finally, we extend our results to Dirac geometry.
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Dissertations / Theses on the topic "Symplectic and Poisson geometry"

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Martino, Maurizio. "Symplectic reflection algebras and Poisson geometry." Thesis, University of Glasgow, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.426614.

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Remsing, Claidiu Cristian. "Tangentially symplectic foliations." Thesis, Rhodes University, 1994. http://hdl.handle.net/10962/d1005233.

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This thesis is concerned principally with tangential geometry and the applications of these concepts to tangentially symplectic foliations. The subject of tangential geometry is still at an elementary stage. The author here systematises current concepts and results and extends them, leading to the definition of vertical connections and vertical G-structures. Tangentially symplectic foliations are then characterised in terms of vertical symplectic forms. Some significant particular cases are discussed.
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Kirchhoff-Lukat, Charlotte Sophie. "Aspects of generalized geometry : branes with boundary, blow-ups, brackets and bundles." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/283007.

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This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.
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Costa, Paulo Henrique Pereira da 1983. "Difusões em variedades de poisson." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306283.

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Orientador: Paulo Regis Caron Ruffino
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
Made available in DSpace on 2018-08-13T23:01:19Z (GMT). No. of bitstreams: 1 Costa_PauloHenriquePereirada_M.pdf: 875708 bytes, checksum: 8862a1813f1bb85b5d0269462a80501e (MD5) Previous issue date: 2009
Resumo: O objetivo desse trabalho é estudar as equações de Hamilton no contexto estocástico. Sendo necessário para tal um pouco de conhecimento a cerca dos seguintes assuntos: cálculo estocástico, geometria de segunda ordem, estruturas simpléticas e de Poisson. Abordamos importantes resultados, dentre eles o teorema de Darboux (coordenadas locais) em variedades simpléticas, teorema de Lie-Weinstein que de certa forma generaliza o teorema de Darboux em variedades de Poisson. Veremos que apesar de o ambiente natural para se estudar sistemas hamiltonianos ser variedades simpléticas, no caso estocástico esses sistemas se adaptam bem em variedades de Poisson. Além disso, para atingir a nossa meta, estudaremos equações diferenciais estocásticas em variedades de dimensão finita usando o operador de Stratonovich
Abstract: This dissertation deals with transfering Hamilton's equations in stochastic context. This requires some knowledge about the following: stochastic calculus, second order geometry and Poisson and simplectic structures. Important results that will be discussed in this theory are Darboux's theorem (local coordinates) for simplectic manifolds, and Lie-Weintein's theorem that is in a certain way of Darboux's theorem on Poisson manifolds. We will see that although the natural environment for studying hamiltonian systems is symplectic manifolds, if we have a Poisson structure we will still be able to study them. Moreover, to achieve our goal, we will study stochastic differential equations on finite dimensional manifolds using the Stratonovich operator
Mestrado
Geometria Estocastica
Mestre em Matemática
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Van, De Ven Christiaan Jozef Farielda. "Quantum Systems and their Classical Limit A C*- Algebraic Approach." Doctoral thesis, Università degli studi di Trento, 2021. http://hdl.handle.net/11572/324358.

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In this thesis we develop a mathematically rigorous framework of the so-called ''classical limit'' of quantum systems and their semi-classical properties. Our methods are based on the theory of strict, also called C*- algebraic deformation quantization. Since this C*-algebraic approach encapsulates both quantum as classical theory in one single framework, it provides, in particular, an excellent setting for studying natural emergent phenomena like spontaneous symmetry breaking (SSB) and phase transitions typically showing up in the classical limit of quantum theories. To this end, several techniques from functional analysis and operator algebras have been exploited and specialised to the context of Schrödinger operators and quantum spin systems. Their semi-classical properties including the possible occurrence of SSB have been investigated and illustrated with various physical models. Furthermore, it has been shown that the application of perturbation theory sheds new light on symmetry breaking in Nature, i.e. in real, hence finite materials. A large number of physically relevant results have been obtained and presented by means of diverse research papers.
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Martin, Shaun K. "Symplectic geometry and gauge theory." Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389209.

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Smith, Ivan. "Symplectic geometry of Lefschetz fibrations." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299234.

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Boalch, Philip Paul. "Symplectic geometry and isomonodromic deformations." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301848.

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at, Andreas Cap@esi ac. "Equivariant Symplectic Geometry of Cotangent Bundles." Moscow Math. J. 1, No.2 (2001) 287-299, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi996.ps.

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Rødland, Lukas. "Symplectic geometry and Calogero-Moser systems." Thesis, Uppsala universitet, Teoretisk fysik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-256742.

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We introduce some basic concepts from symplectic geometry, classical mechanics and integrable systems. We use this theory to show that the rational and the trigonometric Calogero-Moser systems, that is the Hamiltonian systems with Hamiltonian  and  respectively are integrable systems. We do this using symplectic reduction on .
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Books on the topic "Symplectic and Poisson geometry"

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Marsden, Jerrold E., and Tudor S. Ratiu, eds. The Breadth of Symplectic and Poisson Geometry. Boston, MA: Birkhäuser Boston, 2007. http://dx.doi.org/10.1007/b138687.

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V, Karasev M., ed. Coherent transform, quantization and Poisson geometry. Providence, R.I: American Mathematical Society, 1998.

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V, Karasev M., Shishkova Maria, and American Mathematical Society, eds. Quantum algebras and Poisson geometry in mathematical physics. Providence, R.I: American Mathematical Society, 2005.

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Mokhov, O. I. Symplectic and poisson geometry on loop spaces of smooth manifolds and integrable equations. [Amsterdam]: Harwood Academic Publishers, 2001.

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1963-, Shapiro Michael, and Vainshtein Alek 1958-, eds. Cluster algebra and Poisson geometry. Providence, R.I: American Mathematical Society, 2010.

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Puta, Mircea. Hamiltonian mechanical systems and geometric quantization. Dordrecht: Kluwer Academic Publishers, 1993.

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Aebischer, B., M. Borer, M. Kälin, Ch Leuenberger, and H. M. Reimann. Symplectic Geometry. Basel: Birkhäuser Basel, 1994. http://dx.doi.org/10.1007/978-3-0348-7512-7.

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Fomenko, A. T. Symplectic geometry. 2nd ed. Yverdon-les-Bains, Switzerland: Gordon & Breach, 1995.

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Hofer, Helmut, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk, eds. Symplectic Geometry. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-19111-4.

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Fomenko, A. T. Symplectic geometry. New York: Gordon and Breach, 1988.

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Book chapters on the topic "Symplectic and Poisson geometry"

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Koszul, Jean-Louis, and Yi Ming Zou. "Poisson Manifolds." In Introduction to Symplectic Geometry, 91–107. Singapore: Springer Singapore, 2019. http://dx.doi.org/10.1007/978-981-13-3987-5_5.

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Michor, Peter. "Symplectic and Poisson geometry." In Graduate Studies in Mathematics, 411–76. Providence, Rhode Island: American Mathematical Society, 2008. http://dx.doi.org/10.1090/gsm/093/07.

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Medina, Alberto. "Structures de Poisson affines." In Symplectic Geometry and Mathematical Physics, 288–302. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4757-2140-9_14.

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Libermann, Paulette, and Charles-Michel Marle. "Symplectic manifolds and Poisson manifolds." In Symplectic Geometry and Analytical Mechanics, 89–184. Dordrecht: Springer Netherlands, 1987. http://dx.doi.org/10.1007/978-94-009-3807-6_3.

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Wilbour, Don C., and Judith M. Arms. "Reduction Procedures for Poisson Manifolds." In Symplectic Geometry and Mathematical Physics, 462–75. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4757-2140-9_24.

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Vaisman, Izu. "Symplectic Realizations of Poisson Manifolds." In Lectures on the Geometry of Poisson Manifolds, 115–33. Basel: Birkhäuser Basel, 1994. http://dx.doi.org/10.1007/978-3-0348-8495-2_9.

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Huebschmann, Johannes. "On the Quantization of Poisson Algebras." In Symplectic Geometry and Mathematical Physics, 204–33. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4757-2140-9_10.

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Dufour, Jean-Paul. "Formes normales de structures de Poisson." In Symplectic Geometry and Mathematical Physics, 129–35. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4757-2140-9_6.

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Woit, Peter. "The Poisson Bracket and Symplectic Geometry." In Quantum Theory, Groups and Representations, 189–98. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-64612-1_14.

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Ouzilou, R. "Quelques remarques sur les variétés de Poisson-Nijenhuis." In Symplectic Geometry and Mathematical Physics, 355–65. Boston, MA: Birkhäuser Boston, 1991. http://dx.doi.org/10.1007/978-1-4757-2140-9_17.

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Conference papers on the topic "Symplectic and Poisson geometry"

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BANYAGA, AUGUSTIN, and PAUL DONATO. "A NOTE ON ISOTOPIES OF SYMPLECTIC AND POISSON STRUCTURES." In Infinite Dimensional Lie Groups in Geometry and Representation Theory. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812777089_0010.

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Aymerich-Valls, M., and J. C. Marrero. "Coisotropic submanifolds of linear Poisson manifolds and Lagrangian anchored vector subbundles of the symplectic cover." In XX INTERNATIONAL FALL WORKSHOP ON GEOMETRY AND PHYSICS. AIP, 2012. http://dx.doi.org/10.1063/1.4733372.

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Maschke, B. M. J., and A. J. van der Schaft. "Hamiltonian Systems, Pseudo-Poisson Brackets and Their Scattering Representation for Physical Systems." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/vib-8007.

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Abstract This paper is concerned with the definition of the geometric structure of Hamiltonian systems associated with energy–conserving systems in relation with an interconnection topology of their network model. It is also presented how the symplectic structure of standard Hamiltonian systems has to be extended to pseudo–Poisson tensors in order to cope with invariants, equilibria and constraints. Finally a scattering representation of these pseudo–Poisson tensors is defined.
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MCDUFF, DUSA. "WHAT IS SYMPLECTIC GEOMETRY?" In Proceedings of the 13th General Meeting. WORLD SCIENTIFIC, 2009. http://dx.doi.org/10.1142/9789814277686_0002.

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Ishikawa, G., and S. Janeczko. "Bifurcations in symplectic space." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-8.

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ASADI, ESMAEEL, and JAN A. SANDERS. "INTEGRABLE SYSTEMS IN SYMPLECTIC GEOMETRY." In Proceedings of the International Conference on SPT 2007. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812776174_0003.

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Boyom, Michel Nguiffo. "Some lagrangian invariants of symplectic manifolds." In Geometry and Topology of Manifolds. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2007. http://dx.doi.org/10.4064/bc76-0-27.

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Román-Roy, Narciso, Modesto Salgado, Silvia Vilariño, Rui Loja Fernandes, and Roger Picken. "Symmetries in k-Symplectic Field Theories." In GEOMETRY AND PHYSICS: XVI International Fall Workshop. AIP, 2008. http://dx.doi.org/10.1063/1.2958175.

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da Silva, Ana Cannas, Rui Loja Fernandes, and Roger Picken. "4-Manifolds with a Symplectic Bias." In GEOMETRY AND PHYSICS: XVI International Fall Workshop. AIP, 2008. http://dx.doi.org/10.1063/1.2958177.

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WURZBACHER, TILMANN. "INTRODUCTION TO DIFFERENTIABLE MANIFOLDS AND SYMPLECTIC GEOMETRY." In Proceedings of the Summer School. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812810571_0001.

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Reports on the topic "Symplectic and Poisson geometry"

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Blaga, Adara M. Remarks on Poisson Reduction on k-symplectic Manifolds. GIQ, 2012. http://dx.doi.org/10.7546/giq-10-2009-127-132.

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Kalashnikova, Irina. Preconditioner and convergence study for the Quantum Computer Aided Design (QCAD) nonlinear poisson problem posed on the Ottawa Flat 270 design geometry. Office of Scientific and Technical Information (OSTI), May 2012. http://dx.doi.org/10.2172/1044970.

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