DOCSLIB.ORG

Symplectic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Symplectic Geometry Symplectic geometry Daniel Bryan University of Hamburg May 5, 2020 Daniel Bryan (Uni-hamburg) Short title May 5, 2020 1 / 28 Overview 1 Introduction to symplectic geometry Poisson and symplectic manifolds Poisson bracket and Hamiltonian Poisson brackets, coordinates and symplectic form Symplectic form and Hamiltonian 2 Coadjoint Orbits and SU(2) example Coadjoint orbits SU(2) Example Kostant-Kirillov symplectic structure 3 Hamiltonian reduction Locally Hamiltonian actions of Lie algebra Poissonian action of Lie group Daniel Bryan (Uni-hamburg) Short title May 5, 2020 2 / 28 Poisson and symplectic manifolds We consider a phase space M and differentiable functions on them. An algebra of these functions F(M) can be defined. The Poisson bracket can be defined as a bilinear antisymmetric derivation of the algebra : F(M) × F(M) 7! F(M). The Poisson brackets obey the following conditions: Antisymmetry: ff1; f2g = −{f2; f1g ff1; αf2 + βf3g = αff1; f2g + βff1; f3g ff1; f2f3g = ff1; f2gf3 + f2ff1; f3g Leibnitz Rule ff1; ff2; f3ggg + ff3; ff1; f2ggg + ff2; ff1; f3ggg = 0 Jacobi Identity Daniel Bryan (Uni-hamburg) Short title May 5, 2020 3 / 28 Poisson bracket and Hamiltonian One can associate a vector field with the Hamiltonian H of the system XH f = fH; f g. This vector field defines the time evolution of the Hamiltonian f_ = XH f = fH; f g . This means that for conserved quantities fH; f1g = fH; f2g = 0 From the Jacobi identity f1 and f2 are conserved quantities then ff1; f2g is also conserved. A Poisson bracket can be degenerate: this happens when functions f (M) exist such that ff ; gg = 0 for all g. However this is not the case for those defining a symplectic manifold. These functions are known as the center of the Poisson Algebra. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 4 / 28 Poisson brackets, coordinates, and symplectic form In coordinates the Poisson brackets can be written as ij @f1 @f1 ff1; f2g(x) = Σij P (x) @xi @xj using the bilinearity and Leibnitz rule. The function Pij (x) = −Pji (x). is jk ks ij js ik The Jacobi identity becomes P @s P + P @s P + P @s P = 0 Assume M is even dimensional. Then Pij (x) can be invertible meaning that the center of the algebra is trivial such that ij ia −1 bj @s P = −P @s (P )abP . This can be inserted into the Jacobi Triple product where it becomes: P is ja kb −1 −1 −1 s;a;b P P P (@s (P )ab + @b(P )sa + @a(P )bs ) = 0 Pij is invertible this is equivalent to −1 −1 −1 @s (P )ab + @b(P )sa + @a(P )bs = 0 ij i i This can be interpreted as the closed 2-form ! = −Σi<j P dx ^ dx this is d! = 0. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 5 / 28 Poisson manifolds: R3 example A Poisson manifold is a manifold on which a Poisson bracket can be defined. A Poisson manifold is more general than a symplectic manifold as it contains degenerate Poisson brackets from which a symplectic structure cannot be constructed. For example odd dimensional manifolds can be Poisson but are not 3 symplectic. Consider R for which the Poisson brackets are defined as: fxi ; xj g = 2ijk xk fx1; x2g = 2x3; fx1; x3g = −2x2; fx2; x3g = 2x1 Daniel Bryan (Uni-hamburg) Short title May 5, 2020 6 / 28 Symplectic manifolds and Hamiltonian The symplectic manifold (M;!) is defined as a manifold with a non-degenerate closed 2 -form described previously which is globally defined on the manifold. The manifold has a Hamiltonian vector field XH defined as dH = iXH ! or dH = iXH Ω(XH ; •). A symplectic transformation γ is a bijection acting on points m 2 M in the phase space γ : M 7! M. Consider the transformation: ∗ (γ !)m(V ; W ) = !γ(m)(γ∗V ; γ∗W ). Whenever γ∗! = ! the transformation is symplectic. Infinitesimal transformations are specified by the vector field X on M such that LX ! = 0 where LX = d ◦ iX + iX d is the Lie derivative. Poission brackets transform as γff ; gg = fγf ; γgg. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 7 / 28 Symplectic form and Hamiltonian Any Hamiltonian flow is equivalent to a symplectic transformation : Proof This can be seen by considering dH = −iXH !. Now taking the lie derivative w.r.t this vector field we can see that LXH ! = d ◦ (iXH !) + iXH d!. Using d! = 0 and iXH ! = −dH substituting 2 this in the Lie derivative becomes LXH ! = −d H = 0 2n The simplest example of a symplectic manifold is R . This has cannonical coordinates (pi ; qi ) and Poisson brackets fqi ; qj g = 0 fpi ; pj g = 0 and fpi ; qj g = δij . P @H @H The Hamiltonian vector field in this case is X = − @p + @q H i @qi i @pi i Daniel Bryan (Uni-hamburg) Short title May 5, 2020 8 / 28 Coadjoint orbits Coadjoint orbits are important as they are used to find irreducible unitary representations of Lie algebras and physically help to find and quantize the internal degrees of freedom of a physical system. Consider a connected Lie group G and an associated Lie algebra G. The group G acts on X 2 G by the adjoint action X ! (Ad g)(X ) = gXg −1 where g 2 G The co-adjoint action of G on the dual G∗ of the Lie Algebra is: (Ad∗gΞ)(X ) = Ξ(Ad g −1(X )); g 2 G; Ξ 2 G∗; X 2 G These actions can be taken infinitesimally whereby the Lie Algebra acts on itsself and its dual: ad X (Y ) = [X ; Y ]; X ; Y 2 G ad∗ X Ξ(Y ) = −Ξ([X ; Y ]); X ; Y 2 G Ξ 2 G∗ Daniel Bryan (Uni-hamburg) Short title May 5, 2020 9 / 28 SU(2) example Consider the Lie group SU(2), which can be written in terms of the matricies: n α −β∗ o SU(2) = α; β 2 ; jαj2 + jβj2 = 1 β α∗ C Thesu ~ (2) Lie algebra can be written as: n ib c + id o su~ (2) = b; c; d 2 ' 3 −c + id −ib R R Daniel Bryan (Uni-hamburg) Short title May 5, 2020 10 / 28 SU(2) finding coadjoint orbits To find the coadjoint orbit in this case it is important to notice that there is an isomorphism between the Lie algebra and its dual: ∗ 3 F 2 g ' g 2 R . To find the coadjoint orbit of SU(2) one must find the points 0 ∗ 3 0 −1 F 2 g ' g 2 R such that F = gFg where g 2 G. This means we are looking for a conjugation invariant function Q(F ) = Q(F 0) = Q(gFg −1) such that the function Q(F ) = c defines coadjoint orbits F for different constants. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 11 / 28 SU(2) finding coadjoint orbits It is known that the Lie algebrasu ~ (2) is semi-simple and has a trace which can also be used to define an invariant inner product < X ; Y >= tr(XY ). This acts a function tr :su ~ (2) ! C and by considering the cyclicity tr[gFg −1] = tr[F ]. However from the matricies it is obvious that tr[F ] = 0 and hence doesn't give information about coadjoint orbits. However we can try tr[F 2]. This is also conjugation invariant tr[gFg −1gFg −1] = tr[gF 2g −1] = tr[g −1gF 2] = tr[F 2] Daniel Bryan (Uni-hamburg) Short title May 5, 2020 12 / 28 SU(2) orbits Remembering the formsu ~ (2) Lie algebra : n ib c + id o su~ (2) = b; c; d 2 ' 3 −c + id −ib R R Now we take Tr[F 2]: n ib c + id ib c + id o Tr[F 2] = Tr · = −c + id −ib −c + id −ib n −b2 − c2 − d2 ::: o Tr ::: −b2 − c2 − d2 So the trace finally becomes Tr[F 2] = −2(b2 + c2 + d2). Daniel Bryan (Uni-hamburg) Short title May 5, 2020 13 / 28 SU(2) orbits 2 1 2 We can now define r = − 2 Tr[F ] r 2 = b2 + c2 + d2. 2 As b; c; d 2 R we have r 2 R 2 3 Therefore we can interpret r as the radius of a sphere S ⊂ R . 3 Therefore the coadjoint orbits of SU(2) are the set of spheres in R with all possible radii r and the point at the origin r = 0 . Higher order traces Tr[F N ] can be checked and give the same coadjoint orbits so these are all the coadjoint orbits of SU(2). Daniel Bryan (Uni-hamburg) Short title May 5, 2020 14 / 28 Functions on G∗ Consider the space of functions F(G∗) The coadjoint action of G on G∗ induces an action on the functions of G∗. Consider g 2 G and h 2 F(G∗) then Ad∗gh(Ξ) = h(Ad∗g −1(Ξ)) Consider again the function on h(G∗). The differential dh 2 G. This holds because G∗ is a vector space and hence dh is a linear form on G∗ as G∗∗ ∼ G. From this one can also write δΞ 2 G∗ and hence it becomes possible to expand the function as : h(Ξ + δΞ) = h(Ξ + δΞ) = h(Ξ) + δΞ(dh) + O((δΞ)2) Daniel Bryan (Uni-hamburg) Short title May 5, 2020 15 / 28 Kostant -Kirillov symplectic structure The Kostant -Kirillov symplectic structure refers to a symplectic structure on the coadjoint orbit of a Lie Algebra again this is defined by a Poisson bracket and symplectic 2-form.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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............
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • To Be Completed. LECTURE NOTES 1. Hamiltonian
    to be completed. LECTURE NOTES HONGYU HE 1. Hamiltonian Mechanics Let us consider the classical harmonic oscillator mx¨ = −kx (x ∈ R). This is a second order differential equation in terms of Newtonian mechanics. We can convert it into 1st order ordinary differential equations by introducing the momentum p = mx˙; q = x. Then we have p q˙ = , p˙ = −kq. m p The map (q, p) → m i − kqj defines a smooth vector field. The flow curve of this vector field gives the time evolution from any initial state (q(0), p(0)). p2 kq2 Let H(q, p) = 2m + 2 be the Hamiltonian. It represents the energy function. Then ∂H ∂H p = kq, = . ∂q ∂p m So we have ∂H ∂H q˙ = , p˙ = − . ∂p ∂q Generally speaking, for p, q ∈ Rn, for H(q, p) the Hamiltonian, ∂H ∂H q˙ = , p˙ = − . ∂p ∂q is called the Hamiltonian equation. The vector field ∂H ∂H (q, p) → ( , − ) ∂p ∂q 2n is called the Hamiltonian vector field, denoted by XH . In this case, {(q, p)} = R is called the phase space. Any (q, p) is called a state. Any functions on (q, p) is called an observable. In particular H is an observable. Suppose that H is smooth. Then the Hamiltonian equation has local solutions. In other words, 2n for each (q0, p0) ∈ R , there exists > 0 and a function (q(t), p(t))(t ∈ [0, )) such that ∂H ∂H q˙(t) = (q(t), p(t)), p˙(t) = − (q(t), p(t)) ∂p ∂q and q(0) = q0, p(0) = p0. Put φ(q0, p0, t) = (q(t), p(t)).
    [Show full text]
  • 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.
    [Show full text]
  • The Geometry of the Maximum Principle for Affine Connection
    The geometry of the maximum principle for affine connection control systems∗ Andrew D. Lewisy 2000/05/17 Last updated: 2001/01/08 Abstract The maximum principle of Pontryagin is applied to systems where the drift vector field is the geodesic spray corresponding to an affine connection. The result is a second- order differential equation whose right-hand side is the \adjoint Jacobi equation." This latter relates the Hamiltonian geometry of the maximum principle to the affine differen- tial geometry of the system's affine connection. The resulting version of the maximum principle is then be applied to the situation where the cost function is the norm squared of the input force. Keywords. optimal control, mechanical systems, affine connections. AMS Subject Classifications (2020). 49K15, 49Q20, 53B05, 70Q05, 93B27. Contents 1. Introduction2 2. Definitions and notation5 I. Tangent Bundles and Affine Connections7 3. Tangent bundle geometry7 3.1 The tangent lift of a vector field. .7 3.2 The cotangent lift of a vector field. .8 3.3 Joint properties of the tangent and cotangent lift. .9 3.4 The cotangent lift of the vertical lift. 10 3.5 The canonical involution of TTQ......................... 11 3.6 The canonical almost tangent structure. 12 ∗Appears as part of internet supplement to Geometric Control of Mechanical Systems, Francesco Bullo and Andrew D. Lewis, Springer-Verlag, New York-Heidelberg-Berlin, 2004, Number 49 in Texts in Applied Mathematics, ISBN 0-387-22195-6. yProfessor, Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada Email: [email protected], URL: http://www.mast.queensu.ca/~andrew/ Research partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
    [Show full text]
  • Formal (Non)-Commutative Symplectic Geometry
    Formal (non)-commutative symplectic geometry Maxim Kontsevich Some time ago B. Feigin, V. Retakh and I had tried to understand a re- mark of J. Stasheff [15] on open string theory and higher associative algebras [16]. Then I found a strange construction of cohomology classes of mapping class groups using as initial data any differential graded algebra with finite- dimensional cohomology and a kind of Poincar´eduality. Later generalizations to the commutative and Lie cases appeared. In at- tempts to formulate all this I have developed a kind of (non)-commutative cal- culus. The commutative version has fruitful applications in topology of smooth manifolds in dimensions 3. The beginnings of applications are perturbative Chern-Simons theory (S.≥ Axelrod and I.M. Singer [1] and myself), V. Vassiliev’s theory of knot invariants and discriminants (see [19], new results in [2]) and V. Drinfeld’s works on quasi-Hopf algebras (see [6]), also containing elements of Lie calculus. Here I present the formal aspects of the story. Theorem 1.1 is the main motivation for my interest in non-commutative symplectic geometry. Towards the end the exposition becomes a bit more vague and informal. Nevertheless, I hope that I will convince the reader that non-commutative calculus has every right to exist. I have benefited very much from conversations with B. Feigin, V. Retakh, J. Stasheff, R. Bott, D. Kazhdan, G. Segal, I.M. Gelfand, I. Zakharevich, J. Cuntz, Yu. Manin, V. Ginzburg, M. Kapranov and many others. 1 Three infinite-dimensional Lie algebras Let us define three Lie algebras. The first one, denoted by ℓn, is a certain Lie subalgebra of derivations of the free Lie algebra generated by 2n elements p1,...,pn, q1,...,qn.
    [Show full text]
  • On Relative Hamiltonian Diffeomorphisms 3
    On Relative Hamiltonian Diffeomorphisms Ali Sait Demir Istanbul Technical University, Department of Mathematics, 34469 Istanbul, Turkey Abstract. Let Ham(M,L) denote the group of Hamiltonian diffeomorphisms on a symplectic manifold M, leaving a Lagrangian submanifold L ⊂ M invari- ant. In this paper, we show that Ham(M,L) has the fragmentation property, using relative versions of the techniques developed by Thurston and Banyaga. 1. Introduction One of the main concerns in the study of automorphism groups of manifolds is whether the group is simple or perfect. The classical technique of Thurston [7] for showing that the group of C1 diffeomorphisms of a smooth manifold is simple (and hence perfect) requires two main properties of the group: fragmentation and transitivity. Banyaga[1] adaptes these techniques for the group of symplectomor- phisms of a symplectic manifold. In this paper we prove that Ham(M,L): the group of relative Hamiltonian diffeomorphisms on a symplectic manifold M satisfies frag- mentation property. Roughly the group consists of Hamiltonian diffeomorphisms on M that leave a fixed Lagrangian submanifold L ⊂ M invariant, (see Ozan [5] for details). An automorphism group G on a manifold M is called 2-transitive if for all points in M x1; x2; y1; y2 with x1 =6 x2 and y1 =6 y2, there exists diffeomorphisms g1; g2 in G with g1(x1) = y1; g2(x2) = y2 and supp(g1) \ supp(g2) = ;. However Ham(M,L) is far from being transitive since it leaves a submanifold invariant. In- deed this is the main reason for the nonsimplicity of Ham(M,L).
    [Show full text]
  • Symplectic Geometry and Integrable Systems (MATH 538-003) Lecture Notes
    Symplectic Geometry and Integrable Systems (MATH 538-003) Lecture notes Anton Izosimov Last updated: May 2, 2020 Contents 1 Introduction to planar billiards 3 2 Invariant area form of the billiard map and Poincar´e'srecurrence theorem 5 3 Billiards in disks 9 4 Billiards in ellipses 10 5 Geometric meaning of the Joachimsthal integral 13 6 The phase portrait of the billiard in an ellipse 16 7 Liouville integrable maps in 2D 18 8 Proof of the Arnold-Liouville theorem in 2D and Poncelet's closure theorem 20 9 Algebraic integrability of elliptic billiards I 22 10 Algebraic integrability of elliptic billiards II 24 11 Canonical Hamilton's equations 28 12 Symplectic vector spaces and symplectic manifolds 29 13 Vector fields and flows I 32 14 Vector fields and flows II 34 15 Hamiltonian vector fields 37 16 Poisson brackets 42 17 Liouville integrable systems on symplectic manifolds 44 18 Proof of the first part of the Arnold-Liouville theorem: topology 46 1 19 Proof of the second part of the Arnold-Liouville theorem: dynamics 48 20 Noether's principle, first examples of integrable systems 49 21 Further examples of integrable systems 53 22 Introduction to Poisson manifolds 53 23 Symplectic leaves 55 24 Symplectic leaves and Casimir functions 57 25 Lie groups and Lie algebras 59 26 Lie-Poisson brackets and their symplectic leaves 62 27 Lie-Poisson brackets and Casimirs 65 28 Lie-Poisson brackets from Poisson reduction 69 29 Reduction of Hamiltonian dynamics and rigid body motion 71 30 Free rigid body dynamics 74 31 Integrable cases of heavy rigid body motion 76 32 The Toda lattice 78 33 The Adler-Kostant-Symes scheme 80 34 Factorization theorem 84 35 Asymptotic behavior of the Toda lattice 86 2 Lecture 1: Introduction to planar billiards 2 Let D ⊂ R be a compact planar domain bounded by a smooth curve @D.
    [Show full text]