M3/4A16. GEOMETRICAL MECHANICS, Part 1
|
|
- Janel Park
- 6 years ago
- Views:
Transcription
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) May-June 2009 M3/4A16 GEOMETRICAL MECHANICS, Part 1 Setter s signature Checker s signature Editor s signature
2 c 2009 University of London M3/4A16 Page 1 of 5 UNIVERSITY OF LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2009 This paper is also taken for the relevant examination for the Associateship. M3/4A16 GEOMETRICAL MECHANICS, Part 1 Date: Time: Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used.
3 M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 2 of 5 1. The 3D eikonal equation for an optical ray path r(s) R 3 is given by ( d n(r) dr ) = ṙ 2 n ds ds r where ṙ = dr/ds with ṙ = 1. (a) Prove that the 3D eikonal equation preserves ṙ = 1. (b) (c) Explain how the solutions for vectors ṙ and r arrange themselves geometrically, relative to the prescribed gradient n/ r. Derive the 3D eikonal equation from Fermat s principle in the form B 1 0 = δs = δ A 2 n2 (r(τ)) dr dτ dr dτ dτ, with new arclength parameter dτ = nds (optical pathlength). (d) 1. Take the fibre derivative of the Lagrangian to define the canonical momentum; 2. Legendre transform this version of Fermat s principle to determine its Hamiltonian; 3. Write Hamilton s canonical equations for it and 4. Use them to recover the 3D eikonal equation. (e) For L = r p, compute for n = n(r) with r = r. dl { } dτ = L, H
4 M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 3 of 5 2. A steady Euler fluid flow in a rotating frame satisfies u (v dx) + dπ = 0, with π := p u 2 u v, where u is Lie derivative with respect to the divergenceless vector field u = u, with u = 0, and v = u + R, with Coriolis parameter curlr = 2Ω. (a) (b) (c) Write out the Lie derivative relation u (v dx) + dπ = 0 above in two vector forms. In the first form, use the dynamic definition of the Lie-derivative. In the second form, use Cartan s formula in Cartesian coordinates. Explain geometrically what the Cartan version of the steady flow relation means in terms of the vectors u, curl v and (p u 2 ). Show that the steady flow relation u (v dx) + dπ = 0 above implies that the exact two-form defined by dq dp := d(v dx) = curlv ds is invariant under the flow of the divergenceless vector field u. (d) Show that Cartan s formula for the Lie derivative in the steady Euler flow condition implies the Hamiltonian formula and identify the function H. u (curlv ds) = u (dq dp ) = dh (e) (f) Use the result of (2c) to write u Q = u Q and u P = u P in terms of the partial derivatives of H. Write u Q = u Q and u P = u P in terms of a canonical Poisson bracket.
5 M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 4 of 5 3. (a) Compute the Poisson bracket table among the quantities x 1 = 1 2 ( q 2 + p 2 ), x 2 = 1 2 ( p 2 q 2 ), x 3 = p q, with canonical variables (q, p) T R 2. (b) Derive the Poisson bracket for smooth functions on R 3 by changing variables (q, p) T R 2 to (x 1, x 2, x 3 ) R 3 by using the chain rule. Show that the corresponding Hamiltonian vector field for a function H : R 3 R may be expressed as a divergenceless vector field X H = {, H} = S 2 H with S 2 = x 2 1 x 2 2 x (1) Explain why S 2 0 and describe its level set geometrically. (c) Show that the flow of the divergenceless vector field X H preserves volume in 3D. (d) Consider the linear Hamiltonian (whose level sets form planes in R 3 ) H = ax 1 + bx 2 + cx 3 (2) with constant values of (a, b, c). Compute the equation of motion on the space of variables x R 3 obtained by setting d/dt = X H using the Hamiltonian vector field (1) and the linear Hamiltonian (2). (e) Specialise to find the solution of this equation of motion for (a, b, c) = (1, 0, 0).
6 M3/4A16 GEOMETRICAL MECHANICS, Part 1 (2009) Page 5 of 5 4. Symplectic 2 2 time-dependent matrices M i (t) with i = 1, 2, 3 satisfy the relation M i JM T i = J with J = ( ) (a) By using this relation, show that the quantities m i = ṀiM 1 i sp(2) satisfy (Jm i ) T = Jm i for i = 1, 2, 3. (b) The matrices X i = Jm i satisfy X T i = X i for i = 1, 2, 3. Show that J[m i, m j ] = [X i, X j ] J := 2sym(X i JX j ) = X i JX j X j JX i where [m i, m j ] := m i m j m j m i and sym denotes symmetric part. (c) If X = JṀM 1 for derivative Ṁ = M(s, σ)/ s σ=0 and Y = JM M 1 for variational derivative δm = M = M(s, σ)/ σ σ=0, show that equality of cross derivatives in s and σ when evaluated at σ = 0 implies the relation δx := X = Ẏ + [X, Y] J where [X, Y] J := 2sym(XJY). Hint: Define m = ṀM 1 and n = M M 1, subtract cross derivatives m ṅ and then use the result from the previous part. (d) (e) Use the previous relation to compute the Euler-Poincaré equation for evolution resulting from Hamilton s principle ( ) l 0 = δs = δ l(x(s)) ds = tr X δx ds Specialise the Euler-Poincaré equation to the case that l(x) = 1 2 tr(x2 ), where tr denotes trace of a matrix.
[#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 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 informationM3/4A16 Assessed Coursework 1 Darryl Holm Due in class Thursday November 6, 2008 #1 Eikonal equation from Fermat s principle
D. D. Holm November 6, 2008 M3/416 Geom Mech Part 1 1 M3/416 ssessed Coursework 1 Darryl Holm Due in class Thursday November 6, 2008 #1 Eikonal equation from Fermat s principle #1a Prove that the 3D eikonal
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 information1 M3-4-5A16 Assessed Problems # 1: Do all three problems
D. D. Holm M3-4-5A34 Assessed Problems # 1 Due 1 Feb 2013 1 1 M3-4-5A16 Assessed Problems # 1: Do all three problems Exercise 1.1 (Quaternions in Cayley-Klein (CK) parameters). Express all of your answers
More informationClassical Mechanics in Hamiltonian Form
Classical Mechanics in Hamiltonian Form We consider a point particle of mass m, position x moving in a potential V (x). It moves according to Newton s law, mẍ + V (x) = 0 (1) This is the usual and simplest
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 informationM3A23/M4A23. Specimen Paper
UNIVERSITY OF LONDON Course: M3A23/M4A23 Setter: J. Lamb Checker: S. Luzzatto Editor: Editor External: External Date: March 26, 2009 BSc and MSci EXAMINATIONS (MATHEMATICS) May-June 2008 M3A23/M4A23 Specimen
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 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 informationFermat s ray optics. Chapter Fermat s principle
Chapter 1 Fermat s ray optics 1.1 Fermat s principle Fermat s principle states that the path between two points taken by a ray of light is the one traversed in the extremal time. Fermat This principle
More informationFERMAT S RAY OPTICS. Contents
1 FERMAT S RAY OPTICS Contents 1.1 Fermat s principle 3 1.1.1 Three-dimensional eikonal equation 4 1.1.2 Three-dimensional Huygens wave fronts 9 1.1.3 Eikonal equation for axial ray optics 14 1.1.4 The
More informationM2A2 Problem Sheet 3 - Hamiltonian Mechanics
MA Problem Sheet 3 - Hamiltonian Mechanics. The particle in a cone. A particle slides under gravity, inside a smooth circular cone with a vertical axis, z = k x + y. Write down its Lagrangian in a) Cartesian,
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 informationInvariant Lagrangian Systems on Lie Groups
Invariant Lagrangian Systems on Lie Groups Dennis Barrett Geometry and Geometric Control (GGC) Research Group Department of Mathematics (Pure and Applied) Rhodes University, Grahamstown 6140 Eastern Cape
More informationAnalytical Mechanics for Relativity and Quantum Mechanics
Analytical Mechanics for Relativity and Quantum Mechanics Oliver Davis Johns San Francisco State University OXPORD UNIVERSITY PRESS CONTENTS Dedication Preface Acknowledgments v vii ix PART I INTRODUCTION:
More informationVariational principles and Hamiltonian Mechanics
A Primer on Geometric Mechanics Variational principles and Hamiltonian Mechanics Alex L. Castro, PUC Rio de Janeiro Henry O. Jacobs, CMS, Caltech Christian Lessig, CMS, Caltech Alex L. Castro (PUC-Rio)
More informationIn a uniform 3D medium, we have seen that the acoustic Green s function (propagator) is
Chapter Geometrical optics The material in this chapter is not needed for SAR or CT, but it is foundational for seismic imaging. For simplicity, in this chapter we study the variable-wave speed wave equation
More informationHamiltonian flow in phase space and Liouville s theorem (Lecture 5)
Hamiltonian flow in phase space and Liouville s theorem (Lecture 5) January 26, 2016 90/441 Lecture outline We will discuss the Hamiltonian flow in the phase space. This flow represents a time dependent
More informationSeptember 21, :43pm Holm Vol 2 WSPC/Book Trim Size for 9in by 6in
1 GALILEO Contents 1.1 Principle of Galilean relativity 2 1.2 Galilean transformations 3 1.2.1 Admissible force laws for an N-particle system 6 1.3 Subgroups of the Galilean transformations 8 1.3.1 Matrix
More informationSketchy Notes on Lagrangian and Hamiltonian Mechanics
Sketchy Notes on Lagrangian and Hamiltonian Mechanics Robert Jones Generalized Coordinates Suppose we have some physical system, like a free particle, a pendulum suspended from another pendulum, or a field
More informationChapter 1. Principles of Motion in Invariantive Mechanics
Chapter 1 Principles of Motion in Invariantive Mechanics 1.1. The Euler-Lagrange and Hamilton s equations obtained by means of exterior forms Let L = L(q 1,q 2,...,q n, q 1, q 2,..., q n,t) L(q, q,t) (1.1)
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 informationSeminar Geometrical aspects of theoretical mechanics
Seminar Geometrical aspects of theoretical mechanics Topics 1. Manifolds 29.10.12 Gisela Baños-Ros 2. Vector fields 05.11.12 and 12.11.12 Alexander Holm and Matthias Sievers 3. Differential forms 19.11.12,
More informationPart II. Classical Dynamics. Year
Part II Year 28 27 26 25 24 23 22 21 20 2009 2008 2007 2006 2005 28 Paper 1, Section I 8B Derive Hamilton s equations from an action principle. 22 Consider a two-dimensional phase space with the Hamiltonian
More informationModified Equations for Variational Integrators
Modified Equations for Variational Integrators Mats Vermeeren Technische Universität Berlin Groningen December 18, 2018 Mats Vermeeren (TU Berlin) Modified equations for variational integrators December
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 informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationDirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems
Dirac Structures and the Legendre Transformation for Implicit Lagrangian and Hamiltonian Systems Hiroaki Yoshimura Mechanical Engineering, Waseda University Tokyo, Japan Joint Work with Jerrold E. Marsden
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 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 informationLegendre Transforms, Calculus of Varations, and Mechanics Principles
page 437 Appendix C Legendre Transforms, Calculus of Varations, and Mechanics Principles C.1 Legendre Transforms Legendre transforms map functions in a vector space to functions in the dual space. From
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 informationPoincaré (non-holonomic Lagrange) Equations
Department of Theoretical Physics Comenius University Bratislava fecko@fmph.uniba.sk Student Colloqium and School on Mathematical Physics, Stará Lesná, Slovakia, August 23-29, 2010 We will learn: In which
More informationOn explicit integration of two non-holonomic problems
On explicit integration of two non-holonomic problems Alexey V. Borisov 1 1 Institute of Computer Sciences, Izhevsk, Russia Generalized Chaplygin systems Equations of motion of the generalized Chaplygin
More informationHamiltonian aspects of fluid dynamics
Hamiltonian aspects of fluid dynamics CDS 140b Joris Vankerschaver jv@caltech.edu CDS 01/29/08, 01/31/08 Joris Vankerschaver (CDS) Hamiltonian aspects of fluid dynamics 01/29/08, 01/31/08 1 / 34 Outline
More informationG : Statistical Mechanics
G25.2651: Statistical Mechanics Notes for Lecture 1 Defining statistical mechanics: Statistical Mechanics provies the connection between microscopic motion of individual atoms of matter and macroscopically
More informationPHYS 705: Classical Mechanics. Hamiltonian Formulation & Canonical Transformation
1 PHYS 705: Classical Mechanics Hamiltonian Formulation & Canonical Transformation Legendre Transform Let consider the simple case with ust a real value function: F x F x expresses a relationship between
More informationHamiltonian Dynamics from Lie Poisson Brackets
1 Hamiltonian Dynamics from Lie Poisson Brackets Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 12 February 2002 2
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 informationHomework 3. 1 Goldstein Part (a) Theoretical Dynamics September 24, The Hamiltonian is given by
Theoretical Dynamics September 4, 010 Instructor: Dr. Thomas Cohen Homework 3 Submitted by: Vivek Saxena 1 Goldstein 8.1 1.1 Part (a) The Hamiltonian is given by H(q i, p i, t) = p i q i L(q i, q i, t)
More informationPhysics 235 Chapter 7. Chapter 7 Hamilton's Principle - Lagrangian and Hamiltonian Dynamics
Chapter 7 Hamilton's Principle - Lagrangian and Hamiltonian Dynamics Many interesting physics systems describe systems of particles on which many forces are acting. Some of these forces are immediately
More informationHousekeeping. No announcement HW #7: 3.19
1 Housekeeping No announcement HW #7: 3.19 2 Configuration Space vs. Phase Space 1 A given point in configuration space q q prescribes fully the 1,, n configuration of the system at a given time t. The
More informationBalanced models in Geophysical Fluid Dynamics: Hamiltonian formulation, constraints and formal stability
Balanced models in Geophysical Fluid Dynamics: Hamiltonian formulation, constraints and formal stability Onno Bokhove 1 Introduction Most fluid systems, such as the three-dimensional compressible Euler
More informationFINAL EXAM GROUND RULES
PHYSICS 507 Fall 2011 FINAL EXAM Room: ARC-108 Time: Wednesday, December 21, 10am-1pm GROUND RULES There are four problems based on the above-listed material. Closed book Closed notes Partial credit will
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 informationREVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!
Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference
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 informationClassical mechanics of particles and fields
Classical mechanics of particles and fields D.V. Skryabin Department of Physics, University of Bath PACS numbers: The concise and transparent exposition of many topics covered in this unit can be found
More informationBACKGROUND 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 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 informationExercise 1 Classical Bosonic String
Exercise 1 Classical Bosonic String 1. The Relativistic Particle The action describing a free relativistic point particle of mass m moving in a D- dimensional Minkowski spacetime is described by ) 1 S
More informationV (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)
IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common
More informationHamilton-Jacobi theory
Hamilton-Jacobi theory November 9, 04 We conclude with the crowning theorem of Hamiltonian dynamics: a proof that for any Hamiltonian dynamical system there exists a canonical transformation to a set of
More informationHomework 4. Goldstein 9.7. Part (a) Theoretical Dynamics October 01, 2010 (1) P i = F 1. Q i. p i = F 1 (3) q i (5) P i (6)
Theoretical Dynamics October 01, 2010 Instructor: Dr. Thomas Cohen Homework 4 Submitted by: Vivek Saxena Goldstein 9.7 Part (a) F 1 (q, Q, t) F 2 (q, P, t) P i F 1 Q i (1) F 2 (q, P, t) F 1 (q, Q, t) +
More informationNumerical Algorithms as Dynamical Systems
A Study on Numerical Algorithms as Dynamical Systems Moody Chu North Carolina State University What This Study Is About? To recast many numerical algorithms as special dynamical systems, whence to derive
More informationThe Principle of Least Action
The Principle of Least Action Anders Svensson Abstract In this paper, the principle of least action in classical mechanics is studied. The term is used in several different contexts, mainly for Hamilton
More informationLarge-scale atmospheric circulation, semi-geostrophic motion and Lagrangian particle methods
Large-scale atmospheric circulation, semi-geostrophic motion and Lagrangian particle methods Colin Cotter (Imperial College London) & Sebastian Reich (Universität Potsdam) Outline 1. Hydrostatic and semi-geostrophic
More informationLecture 5. Alexey Boyarsky. October 21, Legendre transformation and the Hamilton equations of motion
Lecture 5 Alexey Boyarsky October 1, 015 1 The Hamilton equations of motion 1.1 Legendre transformation and the Hamilton equations of motion First-order equations of motion. In the Lagrangian formulation,
More informationHydrodynamics, Thermodynamics, and Mathematics
Hydrodynamics, Thermodynamics, and Mathematics Hans Christian Öttinger Department of Mat..., ETH Zürich, Switzerland Thermodynamic admissibility and mathematical well-posedness 1. structure of equations
More informationRelativistic Dynamics
Chapter 4 Relativistic Dynamics The most important example of a relativistic particle moving in a potential is a charged particle, say an electron, moving in an electromagnetic field, which might be that
More informationfor changing independent variables. Most simply for a function f(x) the Legendre transformation f(x) B(s) takes the form B(s) = xs f(x) with s = df
Physics 106a, Caltech 1 November, 2018 Lecture 10: Hamiltonian Mechanics I The Hamiltonian In the Hamiltonian formulation of dynamics each second order ODE given by the Euler- Lagrange equation in terms
More informationIntroduction and Vectors Lecture 1
1 Introduction Introduction and Vectors Lecture 1 This is a course on classical Electromagnetism. It is the foundation for more advanced courses in modern physics. All physics of the modern era, from quantum
More informationMaster Algèbre géométrie et théorie des nombres Final exam of differential geometry Lecture notes allowed
Université de Bordeaux U.F. Mathématiques et Interactions Master Algèbre géométrie et théorie des nombres Final exam of differential geometry 2018-2019 Lecture notes allowed Exercise 1 We call H (like
More informationClassical Equations of Motion
3 Classical Equations of Motion Several formulations are in use Newtonian Lagrangian Hamiltonian Advantages of non-newtonian formulations more general, no need for fictitious forces better suited for multiparticle
More informationChaos in Hamiltonian systems
Chaos in Hamiltonian systems Teemu Laakso April 26, 2013 Course material: Chapter 7 from Ott 1993/2002, Chaos in Dynamical Systems, Cambridge http://matriisi.ee.tut.fi/courses/mat-35006 Useful reading:
More informationHamiltonian Field Theory
Hamiltonian Field Theory August 31, 016 1 Introduction So far we have treated classical field theory using Lagrangian and an action principle for Lagrangian. This approach is called Lagrangian field theory
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 informationThe geometry of Euler s equations. lecture 2
The geometry of Euler s equations lecture 2 A Lie group G as a configuration space subgroup of N N matrices for some N G finite dimensional Lie group we can think of it as a matrix group Notation: a,b,
More informationMath 115 ( ) Yum-Tong Siu 1. Canonical Transformation. Recall that the canonical equations (or Hamiltonian equations) = H
Math 115 2006-2007) Yum-Tong Siu 1 ) Canonical Transformation Recall that the canonical equations or Hamiltonian equations) dy = H dx p dp = dx H, where p = F y, and H = F + y p come from the reduction
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 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 informationNonlinear Single-Particle Dynamics in High Energy Accelerators
Nonlinear Single-Particle Dynamics in High Energy Accelerators Part 2: Basic tools and concepts Nonlinear Single-Particle Dynamics in High Energy Accelerators This course consists of eight lectures: 1.
More informationHamiltonian Mechanics
Alain J. Brizard Saint Michael's College Hamiltonian Mechanics 1 Hamiltonian The k second-order Euler-Lagrange equations on con guration space q =(q 1 ; :::; q k ) d @ _q j = @q j ; (1) can be written
More informationLaplace equation in polar coordinates
Laplace equation in polar coordinates The Laplace equation is given by 2 F 2 + 2 F 2 = 0 We have x = r cos θ, y = r sin θ, and also r 2 = x 2 + y 2, tan θ = y/x We have for the partials with respect to
More informationInvariant Variational Problems & Invariant Curve Flows
Invariant Variational Problems & Invariant Curve Flows Peter J. Olver University of Minnesota http://www.math.umn.edu/ olver Oxford, December, 2008 Basic Notation x = (x 1,..., x p ) independent variables
More informationHamiltonian formulation: water waves Lecture 3
Hamiltonian formulation: water waves Lecture 3 Wm.. Hamilton V.E. Zakharov Hamiltonian formulation: water waves This lecture: A. apid review of Hamiltonian machinery (see also extra notes) B. Hamiltonian
More informationPhysics 312, Winter 2007, Practice Final
Physics 312, Winter 2007, Practice Final Time: Two hours Answer one of Question 1 or Question 2 plus one of Question 3 or Question 4 plus one of Question 5 or Question 6. Each question carries equal weight.
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 informationThe Accelerator Hamiltonian in a Straight Coordinate System
Hamiltoninan Dynamics for Particle Accelerators, Lecture 2 The Accelerator Hamiltonian in a Straight Coordinate System Andy Wolski University of Liverpool, and the Cockcroft Institute, Daresbury, UK. Given
More informationLagrangian and Hamiltonian mechanics on Lie algebroids
Lagrangian and Hamiltonian mechanics on Lie algebroids Mechanics and Lie Algebroids Eduardo Martínez University of Zaragoza emf@unizar.es Lisbon, September 11 2007 Abstract I will review the most relevant
More information15. Hamiltonian Mechanics
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationDynamics and Stability application to submerged bodies, vortex streets and vortex-body systems
Dynamics and Stability application to submerged bodies, vortex streets and vortex-body systems Eva Kanso University of Southern California CDS 140B Introduction to Dynamics February 5 and 7, 2008 Fish
More informationDISCRETE VARIATIONAL OPTIMAL CONTROL
DISCRETE VARIATIONAL OPTIMAL CONTROL FERNANDO JIMÉNEZ, MARIN KOBILAROV, AND DAVID MARTÍN DE DIEGO Abstract. This paper develops numerical methods for optimal control of mechanical systems in the Lagrangian
More information1 Hermitian symmetric spaces: examples and basic properties
Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................
More informationFundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics
Fundamentals of Fluid Dynamics: Ideal Flow Theory & Basic Aerodynamics Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI (after: D.J. ACHESON s Elementary Fluid Dynamics ) bluebox.ippt.pan.pl/
More informationKINEMATICS OF CONTINUA
KINEMATICS OF CONTINUA Introduction Deformation of a continuum Configurations of a continuum Deformation mapping Descriptions of motion Material time derivative Velocity and acceleration Transformation
More informationPHY411 Lecture notes Part 2
PHY411 Lecture notes Part 2 Alice Quillen April 6, 2017 Contents 1 Canonical Transformations 2 1.1 Poisson Brackets................................. 2 1.2 Canonical transformations............................
More informationSPECIAL RELATIVITY AND ELECTROMAGNETISM
SPECIAL RELATIVITY AND ELECTROMAGNETISM MATH 460, SECTION 500 The following problems (composed by Professor P.B. Yasskin) will lead you through the construction of the theory of electromagnetism in special
More informationWrite your CANDIDATE NUMBER clearly on each of the THREE answer books provided. Hand in THREE answer books even if they have not all been used.
UNIVERSITY OF LONDON BSc/MSci EXAMINATION May 2007 for Internal Students of Imperial College of Science, Technology and Medicine This paper is also taken for the relevant Examination for the Associateship
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 informationClassical Mechanics. Character: Optative Credits: 12. Type: Theoretical Hours by week: 6. Hours. Theory: 6 Practice: 0
Classical Mechanics Code: 66703 Character: Optative Credits: 12 Type: Theoretical Hours by week: 6 Hours Theory: 6 Practice: 0 General Objective: Provide the student the most important knowledge of classical
More informationBasic hydrodynamics. David Gurarie. 1 Newtonian fluids: Euler and Navier-Stokes equations
Basic hydrodynamics David Gurarie 1 Newtonian fluids: Euler and Navier-Stokes equations The basic hydrodynamic equations in the Eulerian form consist of conservation of mass, momentum and energy. We denote
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 informationMidterm 2 Nov 25, Work alone. Maple is allowed but not on a problem or a part of a problem that says no computer.
Math 416 Name Midterm 2 Nov 25, 2008 Work alone. Maple is allowed but not on a problem or a part of a problem that says no computer. There are 5 problems, do 4 of them including problem 1. Each problem
More informationEquivalence, Invariants, and Symmetry
Equivalence, Invariants, and Symmetry PETER J. OLVER University of Minnesota CAMBRIDGE UNIVERSITY PRESS Contents Preface xi Acknowledgments xv Introduction 1 1. Geometric Foundations 7 Manifolds 7 Functions
More informationIn this section, mathematical description of the motion of fluid elements moving in a flow field is
Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small
More informationRecent Progress in the Integration of Poisson Systems via the Mid Point Rule and Runge Kutta Algorithm
Recent Progress in te Integration of Poisson Systems via te Mid Point Rule and Runge Kutta Algoritm Klaus Bucner, Mircea Craioveanu and Mircea Puta Abstract Some recent progress in te integration of Poisson
More informationPhysics 451/551 Theoretical Mechanics. G. A. Krafft Old Dominion University Jefferson Lab Lecture 18
Physics 451/551 Theoretical Mechanics G. A. Krafft Old Dominion University Jefferson Lab Lecture 18 Theoretical Mechanics Fall 018 Properties of Sound Sound Waves Requires medium for propagation Mainly
More informationPHY 5246: Theoretical Dynamics, Fall September 28 th, 2015 Midterm Exam # 1
Name: SOLUTIONS PHY 5246: Theoretical Dynamics, Fall 2015 September 28 th, 2015 Mierm Exam # 1 Always remember to write full work for what you do. This will help your grade in case of incomplete or wrong
More information