Classical Mechanics and Symplectic Geometry
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Density of Thin Film Billiard Reflection Pseudogroup in Hamiltonian Symplectomorphism Pseudogroup Alexey Glutsyuk
Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk To cite this version: Alexey Glutsyuk. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomor- phism pseudogroup. 2020. hal-03026432v2 HAL Id: hal-03026432 https://hal.archives-ouvertes.fr/hal-03026432v2 Preprint submitted on 6 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk∗yzx December 3, 2020 Abstract Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C1-smooth hypersurface γ ⊂ Rn+1 that is either a global strictly convex closed hypersurface, or a germ of hy- persurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deforma- tions. In the case, when γ is a global convex hypersurface, we show that the latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case, when γ is a germ. -
FLOER HOMOLOGY and MIRROR SYMMETRY I Kenji FUKAYA 0
FLOER HOMOLOGY AND MIRROR SYMMETRY I Kenji FUKAYA Abstract. In this survey article, we explain how the Floer homology of Lagrangian submanifold [Fl1],[Oh1] is related to (homological) mirror symmetry [Ko1],[Ko2]. Our discussion is based mainly on [FKO3]. 0. Introduction. This is the first of the two articles, describing a project in progress to study mirror symmetry and D-brane using Floer homology of Lagrangian submanifold. The tentative goal, which we are far away to achiev, is to prove homological mirror symmetry conjecture by M. Kontsevich (see 3.) The final goal, which is yet very § very far away from us, is to find a new concept of spaces, which is expected in various branches of mathematics and in theoretical physics. Together with several joint authors, I wrote several papers on this project [Fu1], [Fu2], [Fu4], [Fu5], [Fu6], [Fu7], [FKO3], [FOh]. The purpose of this article and part II, is to provide an accesible way to see the present stage of our project. The interested readers may find the detail and rigorous proofs of some of the statements, in those papers. The main purose of Part I is to discribe an outline of our joint paper [FKO3] which is devoted to the obstruction theory to the well-definedness of Floer ho- mology of Lagrangian submanifold. Our emphasis in this article is its relation to mirror symmetry. So we skip most of its application to the geometry of Lagrangian submanifolds. In 1, we review Floer homology of Lagrangian submanifold in the form intro- § duced by Floer and Oh. They assumed various conditions on Lagrangian subman- ifold and symplectic manifold, to define Floer homology. -
Gromov Receives 2009 Abel Prize
Gromov Receives 2009 Abel Prize . The Norwegian Academy of Science Medal (1997), and the Wolf Prize (1993). He is a and Letters has decided to award the foreign member of the U.S. National Academy of Abel Prize for 2009 to the Russian- Sciences and of the American Academy of Arts French mathematician Mikhail L. and Sciences, and a member of the Académie des Gromov for “his revolutionary con- Sciences of France. tributions to geometry”. The Abel Prize recognizes contributions of Citation http://www.abelprisen.no/en/ extraordinary depth and influence Geometry is one of the oldest fields of mathemat- to the mathematical sciences and ics; it has engaged the attention of great mathema- has been awarded annually since ticians through the centuries but has undergone Photo from from Photo 2003. It carries a cash award of revolutionary change during the last fifty years. Mikhail L. Gromov 6,000,000 Norwegian kroner (ap- Mikhail Gromov has led some of the most impor- proximately US$950,000). Gromov tant developments, producing profoundly original will receive the Abel Prize from His Majesty King general ideas, which have resulted in new perspec- Harald at an award ceremony in Oslo, Norway, on tives on geometry and other areas of mathematics. May 19, 2009. Riemannian geometry developed from the study Biographical Sketch of curved surfaces and their higher-dimensional analogues and has found applications, for in- Mikhail Leonidovich Gromov was born on Decem- stance, in the theory of general relativity. Gromov ber 23, 1943, in Boksitogorsk, USSR. He obtained played a decisive role in the creation of modern his master’s degree (1965) and his doctorate (1969) global Riemannian geometry. -
Floer Homology of Lagrangian Submanifolds Are Ex- Pected to Enjoy Can Be Summarized As Follows
FLOER HOMOLOGY OF LAGRANGIAN SUBMANIFOLDS KENJI FUKAYA Abstract. This paper is a survey of Floer theory, which the author has been studying jointly with Y.-G. Oh, H. Ohta, K. Ono. A general idea of the construction is outlined. We also discuss its relation to (homological) mirror symmetry. Especially we describe various conjectures on (homological) mirror symmetry and various partial results towards those conjectures. 1. Introduction This article is a survey of Lagrangian Floer theory, which the author has been studying jointly with Y.-G. Oh, H. Ohta, K. Ono. A major part of our study was published as [28]. Floer homology is invented by A. Floer in 1980’s. There are two areas where Floer homology appears. One is symplectic geometry and the other is topology of 3-4 dimensional manifolds (more specifically the gauge theory). In each of the two areas, there are several different Floer type theories. In symplectic geometry, there are Floer homology of periodic Hamiltonian system ([18]) and Floer homology of Lagrangian submanifolds. Moreover there are two different kinds of Floer theories of contact manifolds ([15, 78]). In gauge theory, there are three kinds of Floer homologies: one based on Yang-Mills theory ([17]), one based on Seiberg-Witten theory ([55, 51]), and Heegard Floer homology ([63]). Those three are closely related to each other. All the Floer type theories have common feature that they define some kinds of homology theory of /2 degree in dimensional space, based on Morse theory. There are many interesting∞ topics∞ to discuss on the general feature of Floer type theories. -
Covariant Hamiltonian Field Theory 3
December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte COVARIANT HAMILTONIAN FIELD THEORY JURGEN¨ STRUCKMEIER and ANDREAS REDELBACH GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH Planckstr. 1, 64291 Darmstadt, Germany and Johann Wolfgang Goethe-Universit¨at Frankfurt am Main Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany [email protected] Received 18 July 2007 Revised 14 December 2020 A consistent, local coordinate formulation of covariant Hamiltonian field theory is pre- sented. Whereas the covariant canonical field equations are equivalent to the Euler- Lagrange field equations, the covariant canonical transformation theory offers more gen- eral means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinites- imal canonical transformation furnishes the most general form of Noether’s theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations. Keywords: Field theory; Hamiltonian density; covariant. PACS numbers: 11.10.Ef, 11.15Kc arXiv:0811.0508v6 [math-ph] 15 Dec 2020 1. Introduction Relativistic field theories and gauge theories are commonly formulated on the basis of a Lagrangian density L1,2,3,4. The space-time evolution of the fields is obtained by integrating the Euler-Lagrange field equations that follow from the four-dimensional representation of Hamilton’s action principle. -
Lecture 1: Basic Concepts, Problems, and Examples
LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introduction to the basic concepts and some of the fundamental problems in symplectic geometry/topology, where along the way various examples are also given for the purpose of illustration. We will often give statements without proofs, which means that their proof is either beyond the scope of this course or will be treated more systematically in later lectures. 1. Symplectic Manifolds Throughout we will assume that M is a C1-smooth manifold without boundary (unless specific mention is made to the contrary). Very often, M will also be closed (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifold M is a 2-form ! 2 Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. d! = 0). (Recall that a 2-form ! 2 Ω2(M) is said to be nondegenerate if for every point p 2 M, !(u; v) = 0 for all u 2 TpM implies v 2 TpM equals 0.) The pair (M; !) is called a symplectic manifold. Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ! is equivalent to the condition that M has an even dimension 2n and the top wedge product !n ≡ ! ^ ! · · · ^ ! is nowhere vanishing on M, i.e., !n is a volume form. In particular, M must be orientable, and is canonically oriented by !n. The nondegeneracy condition is also equivalent to the condition that M is almost complex, i.e., there exists an endomorphism J of TM such that J 2 = −Id. -
Mechanics 84 6.1 2D Particle in Cartesian Coordinates
A Hostetler Handbook v 0.82 Contents Preface v 1 Newton's Laws1 1.1 Many Particles ................................. 2 1.2 Cartesian Coordinates............................. 3 1.3 Polar Coordinates ............................... 4 1.4 Cylindrical Coordinates ............................ 7 1.5 Air Resistance, Friction, and Buoyancy.................... 8 1.6 Charged Particle in a Magnetic Field..................... 17 1.7 Summary: Newton's Laws........................... 20 2 Momentum and Center of Mass 23 2.1 Rocket with no External Force ........................ 24 2.2 Multistage Rockets............................... 25 2.3 Rocket with External Force.......................... 25 2.4 Center of Mass................................. 26 2.5 Angular Momentum of a Single Particle................... 28 2.6 Angular Momentum of a System of Particles ................ 29 2.7 Angular Momentum of a Continuous Mass Distribution .......... 30 2.8 Summary: Momentum and Center of Mass ................. 34 3 Energy 36 3.1 General One-dimensional Systems ...................... 40 3.2 Atwood Machine................................ 43 3.3 Spherically Symmetric Central Forces .................... 44 3.4 The Energy of a System of Particles ..................... 45 3.5 Summary: Energy ............................... 47 4 Oscillations 49 4.1 Simple Harmonic Oscillators.......................... 49 4.2 Damped Harmonic Oscillators......................... 52 4.3 Driven Oscillators ............................... 57 4.4 Summary: Oscillations............................ -
Functor and Natural Operators on Symplectic Manifolds
Mathematical Methods and Techniques in Engineering and Environmental Science Functor and natural operators on symplectic manifolds CONSTANTIN PĂTRĂŞCOIU Faculty of Engineering and Management of Technological Systems, Drobeta Turnu Severin University of Craiova Drobeta Turnu Severin, Str. Călugăreni, No.1 ROMANIA [email protected] http://www.imst.ro Abstract. Symplectic manifolds arise naturally in abstract formulations of classical Mechanics because the phase–spaces in Hamiltonian mechanics is the cotangent bundle of configuration manifolds equipped with a symplectic structure. The natural functors and operators described in this paper can be helpful both for a unified description of specific proprieties of symplectic manifolds and for finding links to various fields of geometric objects with applications in Hamiltonian mechanics. Key-words: Functors, Natural operators, Symplectic manifolds, Complex structure, Hamiltonian. 1. Introduction. diffeomorphism f : M → N , a vector bundle morphism The geometrics objects (like as vectors, covectors, T*f : T*M→ T*N , which covers f , where −1 tensors, metrics, e.t.c.) on a smooth manifold M, x = x :)*)(()*( x * → xf )( * NTMTfTfT are the elements of the total spaces of a vector So, the cotangent functor, )(:* → VBmManT and bundles with base M. The fields of such geometrics objects are the section in corresponding vector more general ΛkT * : Man(m) → VB , are the bundles. bundle functors from Man(m) to VB, where Man(m) By example, the vectors on a smooth manifold M (subcategory of Man) is the category of m- are the elements of total space of vector bundles dimensional smooth manifolds, the morphisms of this (TM ,π M , M); the covectors on M are the elements category are local diffeomorphisms between of total space of vector bundles (T*M , pM , M). -
Aspects of Symplectic Dynamics and Topology: Gauge Anomalies, Chiral Kinetic Theory and Transfer Matrices
ASPECTS OF SYMPLECTIC DYNAMICS AND TOPOLOGY: GAUGE ANOMALIES, CHIRAL KINETIC THEORY AND TRANSFER MATRICES BY VATSAL DWIVEDI DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2017 Urbana, Illinois Doctoral Committee: Associate Professor Shinsei Ryu, Chair Professor Michael Stone, Director of Research Professor James N. Eckstein Professor Jared Bronski Abstract This thesis presents some work on two quite disparate kinds of dynamical systems described by Hamiltonian dynamics. The first part describes a computation of gauge anomalies and their macroscopic effects in a semiclassical picture. The geometric (symplectic) formulation of classical mechanics is used to describe the dynamics of Weyl fermions in even spacetime dimensions, the only quantum input to the symplectic form being the Berry curvature that encodes the spin-momentum locking. The (semi-)classical equations of motion are used in a kinetic theory setup to compute the gauge and singlet currents, whose conservation laws reproduce the nonabelian gauge and singlet anomalies. Anomalous contributions to the hydrodynamic currents for a gas of Weyl fermions at a finite temperature and chemical potential are also calculated, and are in agreement with similar results in literature which were obtained using thermodynamic and/or quantum field theoretical arguments. The second part describes a generalized transfer matrix formalism for noninteracting tight-binding models. The formalism is used to study the bulk and edge spectra, both of which are encoded in the spectrum of the transfer matrices, for some of the common tight-binding models for noninteracting electronic topological phases of matter. -
Constraining Conformal Field Theories with a Higher Spin Symmetry
PUPT-2399 Constraining conformal field theories with a higher spin symmetry Juan Maldacenaa and Alexander Zhiboedovb aSchool of Natural Sciences, Institute for Advanced Study Princeton, NJ, USA bDepartment of Physics, Princeton University Princeton, NJ, USA We study the constraints imposed by the existence of a single higher spin conserved current on a three dimensional conformal field theory. A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. arXiv:1112.1016v1 [hep-th] 5 Dec 2011 This is an extension of the Coleman-Mandula theorem to CFT’s, which do not have a conventional S matrix. We also briefly discuss the case where the higher spin symmetries are “slightly” broken. Contents 1.Introduction ................................. 2 1.1. Organization of the paper . 4 2. Generalities about higher spin currents . 5 1 3. Removing operators in the twist gap, 2 <τ< 1 .................. 7 3.1. Action of the charges on twist one fields . 9 4. Basic facts about three point functions . 10 4.1. General expression for three point functions . 11 5. Argument using bilocal operators . 12 5.1. Light cone limits of correlators of conserved currents . 12 5.2.Gettinganinfinitenumberofcurrents . 15 5.3. Definition of bilocal operators . 17 5.4. Constraining the action of the higher spin charges . 19 5.5. Quantization of N˜: the case of bosons . 24 5.6. -
“Generalized Complex and Holomorphic Poisson Geometry”
“Generalized complex and holomorphic Poisson geometry” Marco Gualtieri (University of Toronto), Ruxandra Moraru (University of Waterloo), Nigel Hitchin (Oxford University), Jacques Hurtubise (McGill University), Henrique Bursztyn (IMPA), Gil Cavalcanti (Utrecht University) Sunday, 11-04-2010 to Friday, 16-04-2010 1 Overview of the Field Generalized complex geometry is a relatively new subject in differential geometry, originating in 2001 with the work of Hitchin on geometries defined by differential forms of mixed degree. It has the particularly inter- esting feature that it interpolates between two very classical areas in geometry: complex algebraic geometry on the one hand, and symplectic geometry on the other hand. As such, it has bearing on some of the most intriguing geometrical problems of the last few decades, namely the suggestion by physicists that a duality of quantum field theories leads to a ”mirror symmetry” between complex and symplectic geometry. Examples of generalized complex manifolds include complex and symplectic manifolds; these are at op- posite extremes of the spectrum of possibilities. Because of this fact, there are many connections between the subject and existing work on complex and symplectic geometry. More intriguing is the fact that complex and symplectic methods often apply, with subtle modifications, to the study of the intermediate cases. Un- like symplectic or complex geometry, the local behaviour of a generalized complex manifold is not uniform. Indeed, its local structure is characterized by a Poisson bracket, whose rank at any given point characterizes the local geometry. For this reason, the study of Poisson structures is central to the understanding of gen- eralized complex manifolds which are neither complex nor symplectic. -
Hamiltonian and Symplectic Symmetries: an Introduction
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 3, July 2017, Pages 383–436 http://dx.doi.org/10.1090/bull/1572 Article electronically published on March 6, 2017 HAMILTONIAN AND SYMPLECTIC SYMMETRIES: AN INTRODUCTION ALVARO´ PELAYO In memory of Professor J.J. Duistermaat (1942–2010) Abstract. Classical mechanical systems are modeled by a symplectic mani- fold (M,ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called sym- plectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton’s equations. Since then a number of connections with combinatorics, finite- dimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent re- sults on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry. 1. Introduction Symplectic geometry is concerned with the study of a notion of signed area, rather than length, distance, or volume. It can be, as we will see, less intuitive than Euclidean or metric geometry and it is taking mathematicians many years to understand its intricacies (which is work in progress). The word “symplectic” goes back to the 1946 book [164] by Hermann Weyl (1885–1955) on classical groups.