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Hamiltonian Systems on Symplectic Manifolds
This is page 147 Printer: Opaque this 5 Hamiltonian Systems on Symplectic Manifolds Now we are ready to geometrize Hamiltonian mechanics to the context of manifolds. First we make phase spaces nonlinear, and then we study Hamiltonian systems in this context. 5.1 Symplectic Manifolds Definition 5.1.1. A symplectic manifold is a pair (P, Ω) where P is a manifold and Ω is a closed (weakly) nondegenerate two-form on P .IfΩ is strongly nondegenerate, we speak of a strong symplectic manifold. As in the linear case, strong nondegeneracy of the two-form Ω means that at each z ∈ P, the bilinear form Ωz : TzP × TzP → R is nondegenerate, that is, Ωz defines an isomorphism → ∗ Ωz : TzP Tz P. Fora(weak) symplectic form, the induced map Ω : X(P ) → X∗(P )be- tween vector fields and one-forms is one-to-one, but in general is not sur- jective. We will see later that Ω is required to be closed, that is, dΩ=0, where d is the exterior derivative, so that the induced Poisson bracket sat- isfies the Jacobi identity and so that the flows of Hamiltonian vector fields will consist of canonical transformations. In coordinates zI on P in the I J finite-dimensional case, if Ω = ΩIJ dz ∧ dz (sum over all I<J), then 148 5. Hamiltonian Systems on Symplectic Manifolds dΩ=0becomes the condition ∂Ω ∂Ω ∂Ω IJ + KI + JK =0. (5.1.1) ∂zK ∂zJ ∂zI Examples (a) Symplectic Vector Spaces. If (Z, Ω) is a symplectic vector space, then it is also a symplectic manifold. -
Time-Dependent Hamiltonian Mechanics on a Locally Conformal
Time-dependent Hamiltonian mechanics on a locally conformal symplectic manifold Orlando Ragnisco†, Cristina Sardón∗, Marcin Zając∗∗ Department of Mathematics and Physics†, Universita degli studi Roma Tre, Largo S. Leonardo Murialdo, 1, 00146 , Rome, Italy. ragnisco@fis.uniroma3.it Department of Applied Mathematics∗, Universidad Polit´ecnica de Madrid. C/ Jos´eGuti´errez Abascal, 2, 28006, Madrid. Spain. [email protected] Department of Mathematical Methods in Physics∗∗, Faculty of Physics. University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland. [email protected] Abstract In this paper we aim at presenting a concise but also comprehensive study of time-dependent (t- dependent) Hamiltonian dynamics on a locally conformal symplectic (lcs) manifold. We present a generalized geometric theory of canonical transformations and formulate a time-dependent geometric Hamilton-Jacobi theory on lcs manifolds. In contrast to previous papers concerning locally conformal symplectic manifolds, here the introduction of the time dependency brings out interesting geometric properties, as it is the introduction of contact geometry in locally symplectic patches. To conclude, we show examples of the applications of our formalism, in particular, we present systems of differential equations with time-dependent parameters, which admit different physical interpretations as we shall point out. arXiv:2104.02636v1 [math-ph] 6 Apr 2021 Contents 1 Introduction 2 2 Fundamentals on time-dependent Hamiltonian systems 5 2.1 Time-dependentsystems. ....... 5 2.2 Canonicaltransformations . ......... 6 2.3 Generating functions of canonical transformations . ................ 8 1 3 Geometry of locally conformal symplectic manifolds 8 3.1 Basics on locally conformal symplectic manifolds . ............... 8 3.2 Locally conformal symplectic structures on cotangent bundles............ -
Conformal Hamiltonian Systems
Journal of Geometry and Physics 39 (2001) 276–300 Conformal Hamiltonian systems Robert McLachlan∗, Matthew Perlmutter IFS, Massey University, Palmerston North, New Zealand 5301 Received 4 August 2000; received in revised form 23 January 2001 Abstract Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called “conformal”. We develop a reduction theory for symmetric con- formal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotan- gent bundles, Lie–Poisson manifolds, and their natural quotients. © 2001 Elsevier Science B.V. All rights reserved. MSC: 53C57; 58705 Sub. Class.: Dynamical systems Keywords: Hamiltonian systems; Conformal Poisson structure 1. Introduction The fields of symplectic geometry and geometric mechanics are closely linked, indeed almost synonymous [10,14]. One of their most important features is that the symplectic diffeomorphisms on a manifold form a group. This property can be taken as the definition of “geometry;” it is natural to try and generalize it. In 1913 Cartan [5,24] found that, subject to certain restrictions, there are just six classes of groups of diffeomorphisms on any manifold, namely (i) Diff itself; (ii) Diffω, the diffeomorphisms preserving a symplectic 2-form ω; (iii) Diffµ, the diffeomorphisms preserving a volume form µ; (iv) Diffα, the diffeomorphisms c preserving a contact form α up to a function; and the conformal groups, (v) Diffω and (vi) c Diffµ, the diffeomorphisms preserving the form ω or µ up to a constant. -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
BRST Charge and Poisson Algebrasy
Discrete Mathematics and Theoretical Computer Science 1, 1997, 115–127 BRST Charge and Poisson Algebrasy H. Caprasse D´epartement d’Astrophysique, Universit´edeLi`ege, Institut de Physique B-5, All´ee du 6 aout, Sart-Tilman B-4000, Liege, Belgium E-Mail: [email protected] An elementary introduction to the classical version of gauge theories is made. The shortcomings of the usual gauge fixing process are pointed out. They justify the need to replace it by a global symmetry: the BRST symmetry and its associated BRST charge. The main mathematical steps required to construct it are described. The algebra of constraints is, in general, a nonlinear Poisson algebra. In the nonlinear case the computation of the BRST charge by hand is hard. It is explained how this computation can be made algorithmic. The main features of a recently created BRST computer algebra program are described. It can handle quadratic algebras very easily. Its capability to compute the BRST charge as a formal power series in the generic case of a cubic algebra is illustrated. Keywords: guage theory, BRST symmetry 1 Introduction Gauge invariance plays a key role in the formulation of all basic physical theories: this is true both at the classical and quantum levels. General relativity as well as all field theoretic models which describe the electromagnetic, the weak and the strong interactions are based on it. One feature of gauge invariant theories is that they contain what physicists call ‘non-observable’ quantities. These are objects which are not in one-to-one correspondance with physical measurements. -
GEOMETRY and STRING THEORY SEMINAR: SPRING 2019 Contents
GEOMETRY AND STRING THEORY SEMINAR: SPRING 2019 ARUN DEBRAY MAY 1, 2019 These notes were taken in UT Austin's geometry and string theory seminar in Spring 2019. I live-TEXed them using vim, and as such there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Rok Gregoric, Qianyu Hao, and Charlie Reid for some helpful comments and corrections. Contents 1. Anomalies and extended conformal manifolds: 1/23/191 2. Integrability and 4D gauge theory I: 1/30/195 3. Integrability and 4D gauge theory II: 2/6/198 4. From Gaussian measures to factorization algebras: 2/13/199 5. Classical free scalar field theory: 2/20/19 11 6. Prefactorization algebras and some examples: 2/27/19 13 7. More examples of prefactorization algebras: 3/6/19 15 8. Free field theories: 3/13/19 17 9. Two kinds of quantization of free theories: 3/27/19 22 10. Abelian Chern-Simons theory 22 11. Holomorphic translation-invariant PFAs and vertex algebras: 4/10/19 24 12. Beta-gamma systems: 4/17/19 25 13. The BV formalism: 4/23/19 27 14. The BV formalism, II: 5/1/19 29 References 31 1. Anomalies and extended conformal manifolds: 1/23/19 These are Arun's prepared notes for his talk, on the paper \Anomalies of duality groups and extended conformal manifolds" [STY18] by Seiberg, Tachikawa, and Yonekura. 1.1. Generalities on anomalies. As we've seen previously in this seminar, if you ask four people what an anomaly in QFT is, you'll probably get four different answers. -
The Hamiltonian Formulation of Geodesics
U.U.D.M. Project Report 2021:36 The Hamiltonian formulation of geodesics Victor Hildebrandsson Examensarbete i matematik, 15 hp Handledare: Georgios Dimitroglou Rizell Examinator: Martin Herschend Juli 2021 Department of Mathematics Uppsala University Abstract We explore the Hamiltonian formulation of the geodesic equation. We start with the definition of a differentiable manifold. Then we continue with studying tensors and differential forms. We define a Riemannian manifold (M; g) and geodesics, the shortest path between two points, on such manifolds. Lastly, we define the symplectic manifold (T ∗M; d(pdq)), and study the connection between geodesics and the flow of Hamilton's equations. 2 Contents 1 Introduction 4 2 Background 5 2.1 Calculus of variations . .5 2.2 Differentiable manifolds . .8 2.3 Tensors and vector bundles . 10 3 Differential forms 13 3.1 Exterior and differential forms . 13 3.2 Integral and exterior derivative of differential forms . 14 4 Riemannian manifolds 17 5 Symplectic manifolds 20 References 25 3 1 Introduction In this thesis, we will study Riemannian manifolds and symplectic manifolds. More specifically, we will study geodesics, and their connection to the Hamilto- nian flow on the symplectic cotangent bundle. A manifold is a topological space which locally looks like Euclidean space. The study of Riemannian manifolds is called Riemannian geometry, the study of symplectic manifolds is called symplectic geometry. Riemannian geometry was first studied by Bernhard Riemann in his habil- itation address "Uber¨ die Hypothesen, welche der Geometrie zugrunde liegen" ("On the Hypotheses on which Geometry is Based"). In particular, it was the first time a mathematician discussed the concept of a differentiable manifold [1]. -
Supersymmetric Extension of Moyal Algebra and Its Application to the Matrix Model
Modern Physics Letters A, c World Scientific Publishing Company f Supersymmetric extension of Moyal algebra and its application to the matrix model Takuya MASUDA Satoru SAITO Department of Physics, Tokyo Metropolitan University Hachioji, Tokyo 192-0397, Japan Received (received date) Revised (revised date) We construct operator representation of Moyal algebra in the presence of fermionic fields. The result is used to describe the matrix model in Moyal formalism, that treat gauge degrees of freedom and outer degrees of freedom equally. 1. Introduction Outer degrees of freedom can be converted into gauge degrees of freedom through compactification. The correspondence between a commutator of matrices and a Poisson bracket of functions used both in M-theory and IIB matrix model implies the fact. When we use the Moyal formalism we can interpolate these two degrees of freedom. There is, however, a problem which we have to overcome before we apply the Moyal formalism to the matrix model. Namely there exists no Moyal formulation of fermionic fields, which is appropriate to describe a supersymmetric theory. Fairlie 1 was the first who wrote the matrix model in Moyal formalism. The supersymmetry, however, has not been fully explored. The main purpose of this article is to construct representation of Moyal algebra to describe the matrix model in Moyal formalism, which treat outer degrees of freedom and the gauge ones equally. Ishibashi et al. have proposed a matrix model which looks like the Green- Schwarz action of type IIB string 2 in the Schild gauge 3 as a constructive definition of string theory 4. This matrix model has the manifest Lorentz invariance and = 2 space-time supersymmetry. -
Odd and Even Poisson Brackets 1
Odd and even Poisson brackets 1 Y. Kosmann-Schwarzbach Centre de Math´ematiques (Unit´eMixte de Recherches du C.N.R.S. 7640) Ecole Polytechnique F-91128 Palaiseau, France Abstract On a Poisson manifold, the divergence of a hamiltonian vector field is a derivation of the algebra of functions, called the modular vector field. In the case of an odd Poisson bracket on a supermanifold, the divergence of a hamilto- nian vector field is a generator of the odd bracket, and it is nontrivial, whether the Poisson structure is symplectic or not. The quantum master equation is a Maurer-Cartan equation written in terms of an odd Poisson bracket (an- tibracket) and of a generator of square 0 of this bracket. The derived bracket of the antibracket with respect to the quantum BRST differential defines the even graded Lie algebra structure of the space of quantum gauge parameters. 1 Introduction In the “geometry of Batalin-Vilkovisky quantization” [13] [14], a supermanifold (M, A) of dimension n|n is equipped with an odd Poisson structure. More precisely, A is a sheaf of Z2-graded Gerstenhaber algebras. A Z2-graded Gerstenhaber algebra A is a Z2-graded commutative, associative algebra, with a bracket, [ , ], satisfying [f,g]= −(−1)(|f|−1)(|g|−1)[g, f] , [f, [g, h]] = [[f,g], h]+(−1)(|f|−1)(|g|−1)[g, [f, h]] , [f,gh] = [f,g]h +(−1)(|f|−1)|g|g[f, h] , 1Quantum Theory and Symmetries, H.-D. Doebner et al.. eds., World Scientific, 2000, 565-571. 1 for all f,g,h ∈ A, where |f| denotes the degree of f. -
Arxiv:1811.07395V2 [Math.SG] 4 Jan 2021 I Lers I Leris Vagba,Lernhr Al Lie-Rinehart Brackets
GRADED POISSON ALGEBRAS ALBERTO S. CATTANEO, DOMENICO FIORENZA, AND RICCARDO LONGONI Abstract. This note is an expanded and updated version of our entry with the same title for the 2006 Encyclopedia of Mathematical Physics. We give a brief overview of graded Poisson algebras, their main properties and their main applications, in the contexts of super differentiable and of derived algebraic geometry. Contents 1. Definitions 2 1.1. Graded vector spaces 2 1.2. Graded algebras and graded Lie algebras 2 1.3. Graded Poisson algebra 2 1.4. Batalin–Vilkovisky algebras 3 2. Examples 4 2.1. Schouten–Nijenhuis bracket 4 2.2. Lie algebroids 5 2.3. Lie algebroid cohomology 6 2.4. Lie–Rinehart algebras 6 2.5. Lie–Rinehart cohomology 7 2.6. Hochschild cohomology 7 2.7. Graded symplectic manifolds 7 2.8. Shifted cotangent bundle 8 2.9. Examples from algebraic topology 8 2.10. ShiftedPoissonstructuresonderivedstacks 8 3. Applications 9 arXiv:1811.07395v2 [math.SG] 4 Jan 2021 3.1. BRST quantization in the Hamiltonian formalism 9 3.2. BV quantization in the Lagrangian formalism 11 4. Related topics 11 4.1. AKSZ 11 4.2. Graded Poisson algebras from cohomology of P∞ 12 References 12 Key words and phrases. Poisson algebras, Poisson manifolds, (graded) symplectic manifolds, Lie algebras, Lie algebroids, BV algebras, Lie-Rinehart algebras, graded manifolds, Schouten– Nijenhuis bracket, Hochschild cohomology, BRST quantization, BV quantization, AKSZ method, derived brackets. A. S. C. acknowledges partial support of SNF Grant No. 200020-107444/1. 1 2 A.S.CATTANEO,D.FIORENZA,ANDR.LONGONI 1. Definitions 1.1. -
Lectures on Poisson Geometry
Rui Loja Fernandes and Ioan Marcut˘ , Lectures on Poisson Geometry March 15, 2014 Springer Foreword These notes grew out of a one semester graduate course taught by us at UIUC in the Spring of 2014. Champaign-Urbana, January 2014 Rui Loja Fernandes and Ioan Marcut˘ , v Preface The aim of these lecture notes is to provide a fast introduction to Poisson geometry. In Classical Mechanics one learns how to describe the time evolution of a me- chanical system with n degrees of freedom: the state of the system at time t is de- n n scribed by a point (q(t); p(t)) in phase space R2 . Here the (q1(t);:::;q (t)) are the configuration coordinates and the (p1(t);:::; pn(t)) are the momentum coordi- nates of the system. The evolution of the system in time is determined by a function n h : R2 ! R, called the hamiltonian: if (q(0); p(0)) is the initial state of the system, then the state at time t is obtained by solving Hamilton’s equations: ( q˙i = ¶H ; ¶ pi (i = 1;:::;n): p˙ = − ¶H ; i ¶qi This description of motion in mechanics is the departing point for Poisson ge- ometry. First, one starts by defining a new product f f ;gg between any two smooth functions f and g, called the Poisson bracket, by setting: n ¶ f ¶g ¶ f ¶g f f ;gg := ∑ i − i : i=1 ¶ pi ¶q ¶q ¶ pi One then observes that, once a function H has been fixed, Hamilton’s equations can be written in the short form: x˙i = fH;xig; (i = 1;:::;2n) i i where x denotes any of the coordinate functions (q ; pi). -
On the AKSZ Formulation of the Poisson Sigma Model
Cattaneo, A S (2001). On the AKSZ formulation of the Poisson sigma model. Letters in Mathematical Physics, 56(2):163-179. Postprint available at: http://www.zora.uzh.ch University of Zurich Posted at the Zurich Open Repository and Archive, University of Zurich. Zurich Open Repository and Archive http://www.zora.uzh.ch Originally published at: Letters in Mathematical Physics 2001, 56(2):163-179. Winterthurerstr. 190 CH-8057 Zurich http://www.zora.uzh.ch Year: 2001 On the AKSZ formulation of the Poisson sigma model Cattaneo, A S Cattaneo, A S (2001). On the AKSZ formulation of the Poisson sigma model. Letters in Mathematical Physics, 56(2):163-179. Postprint available at: http://www.zora.uzh.ch Posted at the Zurich Open Repository and Archive, University of Zurich. http://www.zora.uzh.ch Originally published at: Letters in Mathematical Physics 2001, 56(2):163-179. On the AKSZ formulation of the Poisson sigma model Abstract We review and extend the Alexandrov-Kontsevich-Schwarz-Zaboronsky construction of solutions of the Batalin-Vilkovisky classical master equation. In particular, we study the case of sigma models on manifolds with boundary. We show that a special case of this construction yields the Batalin-Vilkovisky action functional of the Poisson sigma model on a disk. As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds. ON THE AKSZ FORMULATION OF THE POISSON SIGMA MODEL ALBERTO S.