Seminar Geometrical aspects of theoretical mechanics
|
|
- Shonda Griffith
- 6 years ago
- Views:
Transcription
1 Seminar Geometrical aspects of theoretical mechanics Topics 1. Manifolds Gisela Baños-Ros 2. Vector fields and Alexander Holm and Matthias Sievers 3. Differential forms , and Stanislav Katzmin, Enrico Lohmann and Carl Suckfüll 5. Symplectic Geometry and Danilo Uzhogov and Benjamin Schlesinger 6. Hamiltonian Mechanics and Ecaterina Bodnariuc and Richard Busch 7. Application: the force-free rigid body and Richard Busch Time 16:00 17:00 at SR 210, Institute for Theoretical Physics, Brüderstraße 16 Tutor Matthias Schmidt Institute for Theoretical Physics, Brüderstraße 16, room nr. 317B Tel , matthias.schmidtitp.uni-leipzig.de
2
3 1 Manifolds 1 talk of approx. 1 hour Notion of submanifold of R n with examples (e.g. the unit spheres in R 2, R 3 ). Charts and atlases. Motivation: manifolds are necessary for the description of systems with holonomic scleronomic constraints, like the spherical pendulum. Level set theorem: If f : R n R m is differentiable and if y R m is a regular value, then the level set f 1 (y) is a submanifold of R n. Sketch the proof (use the inverse functions theorem). Examples (e.g. the spheres in R 2, R 3, ellipsoids, hyperboloids,...) I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 3 B. T. Bröcker: Analysis, Band II, II.3, II:4
4 2 Vector fields 2 talks of approx. 1 hour Notions of tangent vector and of tangent space T m M of a submanifold M of R n at a point m M. Try to visualize how the tangent space changes from point to point. Construction of the tangent bundle TM = {(x,x) R n R n : x M,X tangent to M at x}. Charts on M induce charts on TM. If M = f 1 (c) is a level set of the mapping f, then T ( f 1 (c) ) = {(x,x) R n R n : f(x) = c,f (x)x = 0}. Give examples, like the unit spheres in R 2, R 3. Notion of tangent mapping (aka differential, derivative) of a differentiable mapping. Definition of vector fields as C -mappings which assign to every point m M a tangent vector X m T m M. Interpretation of vector fields as first order differential operators and thus as derivations of the algebra C (M) of C -functions on M. Commutator of vector fields. Representation of vector fields in coordinates, transformation under a change of coordinates. Notion of integral curve of a vector field, existence and uniqueness, examples. Motivation: In Hamiltonian Mechanics, the time evolution of states is given by the integral curves of a vector field, the Hamiltonian vector field. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 3 B. T. Bröcker: Analysis, Band II, II.3, II:4
5 3 Differential forms 3 talks of approx. 1 hour Algebraic basics Notion of exterior (aka alternating) form on a vector space. The vector spaces k V. exterior product and exterior algebra inner product Definition of the cotangent space T mm of M at m M as the dual space of the tangent space. Cotangent bundle T M and bundle of exterior forms k T M. Differential forms Notion of differential form Examples: Differentials (tangent mappings) df of functions f on M, as well as f 0 df 1 or f 0 df 1 df k. Representation in coordinates, transformation under a change of coordinates exterior product and pull-back exterior derivative de-rham complex, closed and exact forms, Poincaré Lemma Vector analysis in the language of differential forms Riemannian metric and associated Hodge operator Explain the relation between the operators of classical vector analysis grad, rot and div and the exterior derivative of differential forms on R 3. Generalize this to arbitrary 3 dimensional Riemannian manifolds. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 3 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989, Ch. 7 K. Jänich: Vektoranalysis, Springer 1993
6 4 Symplectic Geometry 2 talks of approx. 1 hour symplectic vector spaces Notion of symplectic form on a real vector space existence of a canonical basis symplectic mappings, symplectic group symplectic manifolds Definition and examples Theorem of Darboux (Existence of canonical coordinates) without proof canonical transformations Cotangent bundles Let Q be a manifold. Canonical 1-form θ on M = T Q Coordinates q i on M induce bundle coordinates (q i,p i ) on T M by (x,ξ) (q i (x),p i (x,ξ)), where ξ = p i (x,ξ)dq i. In bundle coordinates, θ = p i dq i. Define ω := dθ. Show that in bundle coordinates, ω = dp i dq i. Conclude that ω is symplectic. Motivation: If a system has configuration space Q, then the associated phase space is given by (T Q,ω). IntheexampleQ = R n,onehast Q = R n R n = R 2n andω = n i=1 dxi dp i. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 7 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989 F. Scheck: Mechanik, Springer 1994 R. Berndt: Einführung in die symplektische Geometrie, Vieweg 1998, , 2.1, 2.2 R. Abraham, J.E. Marsden: Foundations of Mechanics, Addision-Wesley 1978, 3.2, 3.3
7 5 Hamiltonian Mechanics 2 talks of approx. 1 hour mathematical basics Let M be a symplectic manifold. Notion of Hamiltonian vector field X f generated by a function f on M. Representation in canonical coordinates. Poisson bracket {f,g} of functions f,g on M. Under the bracket, the functions form a Lie Algebra and f X f is a Lie algebra homomorphism. The flow of a Hamiltonian vector field is a canonical transformation. physical interpretation Notion of Hamiltonian systems (M, ω, H) Example: M = T Q, where Q = configuration space and H = energy. The points of M are the states of the system: their time evolution is given by the integral curves of the Hamiltonian vector field X H. By writing the equation for the integral curves in terms of canonical coordinates, one obtains the Hamiltonian equations. Interpretation of the functions f : M R as observables, whose value in the state m is given by f(m). Show that the time derivative of the value f ( m(t) ) along the curve m(t) in phase space is given by {f,h} ( m(t) ). Conserved quantities Notion of conserved quantity. Use: the level sets of conserved quantities are invariant under time evolution dynamics reduces to level sets. Show that H is a conserved quantity and that f is conserved iff {H,f} = 0. I. Agricola, T. Friedrich: Globale Analysis, Vieweg 2001, Kap. 7 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989 F. Scheck: Mechanik, Springer 1994 R. Berndt: Einführung in die symplektische Geometrie, Vieweg 1998 R. Abraham, J.E. Marsden: Foundations of Mechanics, Addision-Wesley 1978,
8 6 Application: the force-free rigid body 2 talks of approx. 1 hour Show that after separation of the motion of the center of mass, the configuration space Q is given by the group manifold of the rotation group SO(3). Hence, the phase space is T SO(3). As coordinates, use the Euler angles ϕ, ϑ, ψ and the induced coordinates p ϕ, p ϑ and p ψ on T SO(3). Write down the Lagrange function, perform the Legendre transformation and determine the Hamiltonian H Calculate the Poisson brackets of the components L i of angular momentum and derive the Euler equations Show that H, L 2, L 3 form a system of independent conserved quantities in involution (i.e., the differentials dh, dl 2, dl 3 are linearly independent and their Poisson brackets vanish). Using this example, explain the notion of integrable system: Definition Theorem of Arnold: Dynamics reduces to tori. (if possible, sketch the proof) Visualize the motion on the tori, explain action and angle coordinates and discuss the influence of the ratios of the frequencies. Discuss why one requires the functions to be independent and in involution. F. Scheck: Mechanik, Springer 1994, 3.14, N. Straumann: Klassische Mechanik, Lecture Notes in Physics 289, 11.5 V.I. Arnold: Mathematical Methods of Classical Mechanics, Springer 1989, 49A R. Berndt: Einführung in die symplektische Geometrie, Vieweg 1998 R. Abraham, J.E. Marsden: Foundations of Mechanics, Addision-Wesley 1978,
BACKGROUND IN SYMPLECTIC GEOMETRY
BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationHamiltonian flows, cotangent lifts, and momentum maps
Hamiltonian flows, cotangent lifts, and momentum maps Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Symplectic manifolds Let (M, ω) and (N, η) be symplectic
More informationDierential geometry for Physicists
Dierential geometry for Physicists (What we discussed in the course of lectures) Marián Fecko Comenius University, Bratislava Syllabus of lectures delivered at University of Regensburg in June and July
More informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
More informationA MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS
A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS FRANCESCO BOTTACIN Abstract. In this paper we prove an analogue of the Marsden Weinstein reduction theorem for presymplectic actions of
More informationGauge Fixing and Constrained Dynamics in Numerical Relativity
Gauge Fixing and Constrained Dynamics in Numerical Relativity Jon Allen The Dirac formalism for dealing with constraints in a canonical Hamiltonian formulation is reviewed. Gauge freedom is discussed and
More informationTwisted Poisson manifolds and their almost symplectically complete isotropic realizations
Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Chi-Kwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work
More informationM3-4-5 A16 Notes for Geometric Mechanics: Oct Nov 2011
M3-4-5 A16 Notes for Geometric Mechanics: Oct Nov 2011 Text for the course: Professor Darryl D Holm 25 October 2011 Imperial College London d.holm@ic.ac.uk http://www.ma.ic.ac.uk/~dholm/ Geometric Mechanics
More informationSolutions to the Hamilton-Jacobi equation as Lagrangian submanifolds
Solutions to the Hamilton-Jacobi equation as Lagrangian submanifolds Matias Dahl January 2004 1 Introduction In this essay we shall study the following problem: Suppose is a smooth -manifold, is a function,
More informationThe Geometry of Euler s equation. Introduction
The Geometry of Euler s equation Introduction Part 1 Mechanical systems with constraints, symmetries flexible joint fixed length In principle can be dealt with by applying F=ma, but this can become complicated
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationSolutions of M3-4A16 Assessed Problems # 3 [#1] Exercises in exterior calculus operations
D. D. Holm Solutions to M3-4A16 Assessed Problems # 3 15 Dec 2010 1 Solutions of M3-4A16 Assessed Problems # 3 [#1] Exercises in exterior calculus operations Vector notation for differential basis elements:
More informationCOTANGENT MODELS FOR INTEGRABLE SYSTEMS
COTANGENT MODELS FOR INTEGRABLE SYSTEMS ANNA KIESENHOFER AND EVA MIRANDA Abstract. We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on
More informationHamiltonian Dynamics
Hamiltonian Dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS Feb. 10, 2009 Joris Vankerschaver (CDS) Hamiltonian Dynamics Feb. 10, 2009 1 / 31 Outline 1. Introductory concepts; 2. Poisson brackets;
More informationTHE LAGRANGIAN AND HAMILTONIAN MECHANICAL SYSTEMS
THE LAGRANGIAN AND HAMILTONIAN MECHANICAL SYSTEMS ALEXANDER TOLISH Abstract. Newton s Laws of Motion, which equate forces with the timerates of change of momenta, are a convenient way to describe mechanical
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationLECTURE 1: LINEAR SYMPLECTIC GEOMETRY
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 *********************************************************************************
More informationThe Kepler problem from a differential geometry point of view
The Kepler problem from a differential geometry point of view 1 The Kepler problem from a differential geometry point of view Translation of the original German Version: Das Keplerproblem differentialgeometrisch
More informationA little taste of symplectic geometry
A little taste of symplectic geometry Timothy Goldberg Thursday, October 4, 2007 Olivetti Club Talk Cornell University 1 2 What is symplectic geometry? Symplectic geometry is the study of the geometry
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More information[#1] R 3 bracket for the spherical pendulum
.. Holm Tuesday 11 January 2011 Solutions to MSc Enhanced Coursework for MA16 1 M3/4A16 MSc Enhanced Coursework arryl Holm Solutions Tuesday 11 January 2011 [#1] R 3 bracket for the spherical pendulum
More informationPreface On August 15, 2012, I received the following email message from Ryan Budney (ryan.budney@gmail.com). Hi Dick, Here s an MO post that s right up your alley. :) http://mathoverflow.net/questions/104750/about-a-letter-by-richard-palais-of-1965
More informationSymplectic and Poisson Manifolds
Symplectic and Poisson Manifolds Harry Smith In this survey we look at the basic definitions relating to symplectic manifolds and Poisson manifolds and consider different examples of these. We go on to
More informationSome aspects of the Geodesic flow
Some aspects of the Geodesic flow Pablo C. Abstract This is a presentation for Yael s course on Symplectic Geometry. We discuss here the context in which the geodesic flow can be understood using techniques
More informationREMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF HAMILTONIAN SYSTEMS. Eduardo D. Sontag. SYCON - Rutgers Center for Systems and Control
REMARKS ON THE TIME-OPTIMAL CONTROL OF A CLASS OF HAMILTONIAN SYSTEMS Eduardo D. Sontag SYCON - Rutgers Center for Systems and Control Department of Mathematics, Rutgers University, New Brunswick, NJ 08903
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationEva Miranda. UPC-Barcelona and BGSMath. XXV International Fall Workshop on Geometry and Physics Madrid
b-symplectic manifolds: going to infinity and coming back Eva Miranda UPC-Barcelona and BGSMath XXV International Fall Workshop on Geometry and Physics Madrid Eva Miranda (UPC) b-symplectic manifolds Semptember,
More informationLIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES
LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES BENJAMIN HOFFMAN 1. Outline Lie algebroids are the infinitesimal counterpart of Lie groupoids, which generalize how we can talk about symmetries
More informationGeometric Quantization
math-ph/0208008 Geometric Quantization arxiv:math-ph/0208008v3 4 Sep 2002 William Gordon Ritter Jefferson Physical Laboratory, Harvard University Cambridge, MA 02138, USA February 3, 2008 Abstract We review
More informationBRST 2006 (jmf) 7. g X (M) X ξ X. X η = [ξ X,η]. (X θ)(η) := X θ(η) θ(x η) = ξ X θ(η) θ([ξ X,η]).
BRST 2006 (jmf) 7 Lecture 2: Symplectic reduction In this lecture we discuss group actions on symplectic manifolds and symplectic reduction. We start with some generalities about group actions on manifolds.
More informationEXERCISES IN POISSON GEOMETRY
EXERCISES IN POISSON GEOMETRY The suggested problems for the exercise sessions #1 and #2 are marked with an asterisk. The material from the last section will be discussed in lecture IV, but it s possible
More informationGEOMETRICAL TOOLS FOR PDES ON A SURFACE WITH ROTATING SHALLOW WATER EQUATIONS ON A SPHERE
GEOETRICAL TOOLS FOR PDES ON A SURFACE WITH ROTATING SHALLOW WATER EQUATIONS ON A SPHERE BIN CHENG Abstract. This is an excerpt from my paper with A. ahalov [1]. PDE theories with Riemannian geometry are
More informationReminder on basic differential geometry
Reminder on basic differential geometry for the mastermath course of 2013 Charts Manifolds will be denoted by M, N etc. One should think of a manifold as made out of points (while the elements of a vector
More informationLecture I: Constrained Hamiltonian systems
Lecture I: Constrained Hamiltonian systems (Courses in canonical gravity) Yaser Tavakoli December 15, 2014 1 Introduction In canonical formulation of general relativity, geometry of space-time is given
More informationGeodesic Equivalence in sub-riemannian Geometry
03/27/14 Equivalence in sub-riemannian Geometry Supervisor: Dr.Igor Zelenko Texas A&M University, Mathematics Some Preliminaries: Riemannian Metrics Let M be a n-dimensional surface in R N Some Preliminaries:
More informationGlobal Formulations of Lagrangian and Hamiltonian Dynamics on Embedded Manifolds
1 Global Formulations of Lagrangian and Hamiltonian Dynamics on Embedded Manifolds By Taeyoung Lee, Melvin Leok, and N. Harris McClamroch Mechanical and Aerospace Engineering, George Washington University,
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationAn Invitation to Geometric Quantization
An Invitation to Geometric Quantization Alex Fok Department of Mathematics, Cornell University April 2012 What is quantization? Quantization is a process of associating a classical mechanical system to
More informationNon-associative Deformations of Geometry in Double Field Theory
Non-associative Deformations of Geometry in Double Field Theory Michael Fuchs Workshop Frontiers in String Phenomenology based on JHEP 04(2014)141 or arxiv:1312.0719 by R. Blumenhagen, MF, F. Haßler, D.
More informationShifted Symplectic Derived Algebraic Geometry and generalizations of Donaldson Thomas Theory
Shifted Symplectic Derived Algebraic Geometry and generalizations of Donaldson Thomas Theory Lecture 2 of 3:. D-critical loci and perverse sheaves Dominic Joyce, Oxford University KIAS, Seoul, July 2018
More informationNoether Symmetries and Conserved Momenta of Dirac Equation in Presymplectic Dynamics
International Mathematical Forum, 2, 2007, no. 45, 2207-2220 Noether Symmetries and Conserved Momenta of Dirac Equation in Presymplectic Dynamics Renato Grassini Department of Mathematics and Applications
More informationSymplectic geometry of deformation spaces
July 15, 2010 Outline What is a symplectic structure? What is a symplectic structure? Denition A symplectic structure on a (smooth) manifold M is a closed nondegenerate 2-form ω. Examples Darboux coordinates
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationAn Introduction to Symplectic Geometry
An Introduction to Symplectic Geometry Alessandro Fasse Institute for Theoretical Physics University of Cologne These notes are a short sum up about two talks that I gave in August and September 2015 an
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationHamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics. Manuel de León Institute of Mathematical Sciences CSIC, Spain
Hamilton-Jacobi theory on Lie algebroids: Applications to nonholonomic mechanics Manuel de León Institute of Mathematical Sciences CSIC, Spain joint work with J.C. Marrero (University of La Laguna) D.
More informationDiscrete Dirac Mechanics and Discrete Dirac Geometry
Discrete Dirac Mechanics and Discrete Dirac Geometry Melvin Leok Mathematics, University of California, San Diego Joint work with Anthony Bloch and Tomoki Ohsawa Geometric Numerical Integration Workshop,
More informationVortex Equations on Riemannian Surfaces
Vortex Equations on Riemannian Surfaces Amanda Hood, Khoa Nguyen, Joseph Shao Advisor: Chris Woodward Vortex Equations on Riemannian Surfaces p.1/36 Introduction Yang Mills equations: originated from electromagnetism
More informationR-flux string sigma model and algebroid duality on Lie 3-algebroids
R-flux string sigma model and algebroid duality on Lie 3-algebroids Marc Andre Heller Tohoku University Based on joint work with Taiki Bessho (Tohoku University), Noriaki Ikeda (Ritsumeikan University)
More information4 Riemannian geometry
Classnotes for Introduction to Differential Geometry. Matthias Kawski. April 18, 2003 83 4 Riemannian geometry 4.1 Introduction A natural first step towards a general concept of curvature is to develop
More informationA CRASH COURSE IN EULER-POINCARÉ REDUCTION
A CRASH COURSE IN EULER-POINCARÉ REDUCTION HENRY O. JACOBS Abstract. The following are lecture notes from lectures given at the Fields Institute during the Legacy of Jerrold E. Marsden workshop. In these
More informationA DIFFERENTIAL GEOMETRIC APPROACH TO FLUID MECHANICS
International Journal of Scientific and Research Publications Volume 5 Issue 9 September 2015 1 A DIFFERENTIAL GEOMETRIC APPROACH TO FLUID MECHANICS Mansour Hassan Mansour * M A Bashir ** * Department
More informationCocycles and stream functions in quasigeostrophic motion
Journal of Nonlinear Mathematical Physics Volume 15, Number 2 (2008), 140 146 Letter Cocycles and stream functions in quasigeostrophic motion Cornelia VIZMAN West University of Timişoara, Romania E-mail:
More informationTwisted geodesic flows and symplectic topology
Mike Usher UGA September 12, 2008/ UGA Geometry Seminar Outline 1 Classical mechanics of a particle in a potential on a manifold Lagrangian formulation Hamiltonian formulation 2 Magnetic flows Lagrangian
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationContact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples
Contact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples Vladimir Rubtsov, ITEP,Moscow and LAREMA, Université d Angers Workshop "Geometry and Fluids" Clay Mathematical Institute,
More informationHAPPY BIRTHDAY TONY! Chicago Blochfest June 30, 2105
HAPPY BIRTHDAY TONY! 1 CLEBSCH OPTIMAL CONTROL Tudor S. Ratiu Section de Mathématiques Ecole Polytechnique Fédérale de Lausanne, Switzerland tudor.ratiu@epfl.ch 2 PLAN OF THE PRESENTATION Symmetric representation
More informationLECTURE 4: SYMPLECTIC GROUP ACTIONS
LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 -action on (M, ω) is
More informationPatrick Iglesias-Zemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick Iglesias-Zemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationMaster of Science in Advanced Mathematics and Mathematical Engineering
Master of Science in Advanced Mathematics and Mathematical Engineering Title: Constraint algorithm for singular k-cosymplectic field theories Author: Xavier Rivas Guijarro Advisor: Francesc Xavier Gràcia
More informationSymplectic Geometry versus Riemannian Geometry
Symplectic Geometry versus Riemannian Geometry Inês Cruz, CMUP 2010/10/20 - seminar of PIUDM 1 necessarily very incomplete The aim of this talk is to give an overview of Symplectic Geometry (there is no
More informationManifolds in Fluid Dynamics
Manifolds in Fluid Dynamics Justin Ryan 25 April 2011 1 Preliminary Remarks In studying fluid dynamics it is useful to employ two different perspectives of a fluid flowing through a domain D. The Eulerian
More informationDeformations of coisotropic submanifolds in symplectic geometry
Deformations of coisotropic submanifolds in symplectic geometry Marco Zambon IAP annual meeting 2015 Symplectic manifolds Definition Let M be a manifold. A symplectic form is a two-form ω Ω 2 (M) which
More informationGlobal Properties of Integrable Hamiltonian Systems
ISSN 1560-3547, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 6, pp. 588 630. c Pleiades Publishing, Ltd., 2008. JÜRGEN MOSER 80 Global Properties of Integrable Hamiltonian Systems O. Lukina *, F. Takens,
More informationGeometry and Dynamics of singular symplectic manifolds. Session 9: Some applications of the path method in b-symplectic geometry
Geometry and Dynamics of singular symplectic manifolds Session 9: Some applications of the path method in b-symplectic geometry Eva Miranda (UPC-CEREMADE-IMCCE-IMJ) Fondation Sciences Mathématiques de
More informationfy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))
1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical
More informationPhysics 5153 Classical Mechanics. Canonical Transformations-1
1 Introduction Physics 5153 Classical Mechanics Canonical Transformations The choice of generalized coordinates used to describe a physical system is completely arbitrary, but the Lagrangian is invariant
More informationFoundation Modules MSc Mathematics. Winter Term 2018/19
F4A1-V3A2 Algebra II Prof. Dr. Catharina Stroppel The first part of the course will start from linear group actions and study some invariant theory questions with several applications. We will learn basic
More informationPREQUANTIZATION OF SYMPLECTIC SUPERMANIFOLDS
Ninth International Conference on Geometry, Integrability and Quantization June 8 13, 2007, Varna, Bulgaria Ivaïlo M. Mladenov, Editor SOFTEX, Sofia 2008, pp 301 307 Geometry, Integrability and Quantization
More informationGODBILLON-VEY CLASSES OF SYMPLECTIC FOLIATIONS. Kentaro Mikami
PACIFIC JOURNAL OF MATHEMATICS Vol. 194, No. 1, 2000 GODBILLON-VEY CLASSES OF SYMPLECTIC FOLIATIONS Kentaro Mikami Dedicated to the memory of Professor Shukichi Tanno Each transversally oriented foliation
More informationStability Subject to Dynamical Accessibility
Stability Subject to Dynamical Accessibility P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin morrison@physics.utexas.edu http://www.ph.utexas.edu/
More informationSymmetries in Non Commutative Configuration Space
Symmetries in Non Commutative Configuration Space Univ. Federal do Rio de Janeiro (UFRJ), Brazil E-mail: vanhecke@if.ufrj.br Antonio Roberto da Silva Univ. Federal do Rio de Janeiro (UFRJ), Brazil E-mail:
More informationLeft invariant geometry of Lie groups by Patrick Eberlein August Table of Contents
1 Left invariant geometry of Lie groups by Patrick Eberlein August 2002 Table of Contents Introduction Section 1 Basic properties and examples of symplectic structures (1.1) Lie derivative and exterior
More informationTime-optimal control of a 3-level quantum system and its generalization to an n-level system
Proceedings of the 7 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 7 Time-optimal control of a 3-level quantum system and its generalization to an n-level
More informationNotes on symplectic geometry and group actions
Notes on symplectic geometry and group actions Peter Hochs November 5, 2013 Contents 1 Example of Hamiltonian mechanics 1 2 Proper actions and Hausdorff quotients 4 3 N particles in R 3 7 4 Commuting actions
More informationThe Toda Lattice. Chris Elliott. April 9 th, 2014
The Toda Lattice Chris Elliott April 9 th, 2014 In this talk I ll introduce classical integrable systems, and explain how they can arise from the data of solutions to the classical Yang-Baxter equation.
More informationTowards Discrete Exterior Calculus and Discrete Mechanics for Numerical Relativity
Towards Discrete Exterior Calculus and Discrete Mechanics for Numerical Relativity Melvin Leok Mathematics, University of Michigan, Ann Arbor. Joint work with Mathieu Desbrun, Anil Hirani, and Jerrold
More informationPoisson geometry of b-manifolds. Eva Miranda
Poisson geometry of b-manifolds Eva Miranda UPC-Barcelona Rio de Janeiro, July 26, 2010 Eva Miranda (UPC) Poisson 2010 July 26, 2010 1 / 45 Outline 1 Motivation 2 Classification of b-poisson manifolds
More informationHigher order Koszul brackets
Higher order Koszul brackets Hovhannes Khudaverdian University of anchester, anchester, UK XXXY WORKSHOP ON GEOETRIC ETHODS IN PHYSICS 26 June-2 July, Bialoweza, Poland The talk is based on the work with
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationM3/4A16. GEOMETRICAL MECHANICS, Part 1
M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 1 of 5 UNIVERSITY OF LONDON Course: M3/4A16 Setter: Holm Checker: Gibbons Editor: Chen External: Date: January 27, 2008 BSc and MSci EXAMINATIONS (MATHEMATICS)
More informationCurves in the configuration space Q or in the velocity phase space Ω satisfying the Euler-Lagrange (EL) equations,
Physics 6010, Fall 2010 Hamiltonian Formalism: Hamilton s equations. Conservation laws. Reduction. Poisson Brackets. Relevant Sections in Text: 8.1 8.3, 9.5 The Hamiltonian Formalism We now return to formal
More informationLie algebra cohomology
Lie algebra cohomology Relation to the de Rham cohomology of Lie groups Presented by: Gazmend Mavraj (Master Mathematics and Diploma Physics) Supervisor: J-Prof. Dr. Christoph Wockel (Section Algebra and
More informationAction-angle coordinates and geometric quantization. Eva Miranda. Barcelona (EPSEB,UPC) STS Integrable Systems
Action-angle coordinates and geometric quantization Eva Miranda Barcelona (EPSEB,UPC) STS Integrable Systems Eva Miranda (UPC) 6ecm July 3, 2012 1 / 30 Outline 1 Quantization: The general picture 2 Bohr-Sommerfeld
More informationSynchro-Betatron Motion in Circular Accelerators
Outlines Synchro-Betatron Motion in Circular Accelerators Kevin Li March 30, 2011 Kevin Li Synchro-Betatron Motion 1/ 70 Outline of Part I Outlines Part I: and Model Introduction Part II: The Transverse
More informationGeometric structures and Pfaffian groupoids Geometry and Algebra of PDEs - Universitetet i Tromsø
Geometric structures and Pfaffian groupoids Geometry and Algebra of PDEs - Universitetet i Tromsø (joint work with Marius Crainic) 6 June 2017 Index 1 G-structures 2 Γ-structures 3 Pfaffian objects 4 Almost
More informationPhysics 106b: Lecture 7 25 January, 2018
Physics 106b: Lecture 7 25 January, 2018 Hamiltonian Chaos: Introduction Integrable Systems We start with systems that do not exhibit chaos, but instead have simple periodic motion (like the SHO) with
More informationThe Atiyah bundle and connections on a principal bundle
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata
More informationLet V, W be two finite-dimensional vector spaces over R. We are going to define a new vector space V W with two properties:
5 Tensor products We have so far encountered vector fields and the derivatives of smooth functions as analytical objects on manifolds. These are examples of a general class of objects called tensors which
More informationHybrid Routhian Reduction of Lagrangian Hybrid Systems
Hybrid Routhian Reduction of Lagrangian Hybrid Systems Aaron D. Ames and Shankar Sastry Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, CA 94720
More informationSurvey on exterior algebra and differential forms
Survey on exterior algebra and differential forms Daniel Grieser 16. Mai 2013 Inhaltsverzeichnis 1 Exterior algebra for a vector space 1 1.1 Alternating forms, wedge and interior product.....................
More informationRelation of the covariant and Lie derivatives and its application to Hydrodynamics
Relation of the covariant and Lie derivatives and its application to Hydrodynamics Yang Cao TU Darmstadt March 12, 2010 The Euler hydrodynamic equation on manifolds Let M denote a compact oriented Riemannian
More informationEva Miranda. UPC-Barcelona. (joint with Victor Guillemin and Ana Rita Pires) Zaragoza, February
From b-poisson manifolds to symplectic mapping torus and back Eva Miranda UPC-Barcelona (joint with Victor Guillemin and Ana Rita Pires) Zaragoza, February 8 2011 Eva Miranda (UPC) Poisson Day February
More informationPoisson Manifolds Bihamiltonian Manifolds Bihamiltonian systems as Integrable systems Bihamiltonian structure as tool to find solutions
The Bi hamiltonian Approach to Integrable Systems Paolo Casati Szeged 27 November 2014 1 Poisson Manifolds 2 Bihamiltonian Manifolds 3 Bihamiltonian systems as Integrable systems 4 Bihamiltonian structure
More information[#1] Exercises in exterior calculus operations
D. D. Holm M3-4A16 Assessed Problems # 3 Due when class starts 13 Dec 2012 1 M3-4A16 Assessed Problems # 3 Do all four problems [#1] Exercises in exterior calculus operations Vector notation for differential
More informationConvex Symplectic Manifolds
Convex Symplectic Manifolds Jie Min April 16, 2017 Abstract This note for my talk in student symplectic topology seminar in UMN is an gentle introduction to the concept of convex symplectic manifolds.
More informationLECTURE 26: THE CHERN-WEIL THEORY
LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if
More informationThe topology of symplectic four-manifolds
The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form
More information