DOCSLIB.ORG

Analytical Mechanics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Analytical Mechanics Analytical Mechanics Analytical mechanics is the foundation of many areas of theoretical physics, including quantum theory and statistical mechanics, and has wide-ranging applications in engineer- ing and celestial mechanics. This introduction to the basic principles and methods of analytical mechanics covers Lagrangian and Hamiltonian dynamics, rigid bodies, small oscillations, canonical transformations and Hamilton-Jacobi theory. This fully up-to-date textbook includes detailed mathematical appendices and addresses a number of advanced topics, some of them of a geometric or topological character. These include Bertrand’s theorem, proof that action is least, spontaneous symmetry breakdown, constrained Hamil- tonian systems, non-integrability criteria, KAM theory, classical field theory, Lyapunov functions, geometric phases and Poisson manifolds. Providing worked examples, end-of- chapter problems and discussion of ongoing research in the field, it is suitable for advanced undergraduate students and graduate students studying analytical mechanics. Nivaldo A. Lemos is Associate Professor of Physics at Federal Fluminense University, Brazil. He was previously a visiting scholar at the Massachusetts Institute of Technology. His main research areas are quantum cosmology, quantum field theory and the teaching of classical mechanics. Analytical Mechanics NIVALDO A. LEMOS Fluminense Federal University University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108416580 DOI: 10.1017/9781108241489 © 2004, Editora Livraria da F´ısica, Sao˜ Paulo English translation © Cambridge University Press 2018 Revised and enlarged translation of Mecanicaˆ Anal´ıtica by Nivaldo A. Lemos originally published in Portuguese by Editora Livraria da F´ısica, Sao˜ Paulo 2004 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. English edition first published 2018 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Lemos, Nivaldo A., 1952– author. Title: Analytical mechanics / Nivaldo A. Lemos, Fluminense Federal University. Other titles: Mecanicaˆ anal´ıtica. English Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2018. | Includes bibliographical references and index. Identifiers: LCCN 2018011108| ISBN 9781108416580 (hb) | ISBN 1108416586 (hb) Subjects: LCSH: Mechanics, Analytic. Classification: LCC QA805 .L4313 2018 | DDC 531.01/515–dc23 LC record available at https://lccn.loc.gov/2018011108 ISBN 978-1-108-41658-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. To my wife Marcia and my daughters Cintia, Luiza and Beatriz Contents Preface page xi 1 Lagrangian Dynamics 1 1.1 Principles of Newtonian Mechanics 1 1.2 Constraints 7 1.3 Virtual Displacements and d’Alembert’s Principle 11 1.4 Generalised Coordinates and Lagrange’s Equations 17 1.5 Applications of Lagrange’s Equations 24 1.6 Generalised Potentials and Dissipation Function 28 1.7 Central Forces and Bertrand’s Theorem 31 Problems 37 2 Hamilton’s Variational Principle 42 2.1 Rudiments of the Calculus of Variations 42 2.2 Variational Notation 49 2.3 Hamilton’s Principle and Lagrange’s Equations 50 2.4 Hamilton’s Principle in the Non-Holonomic Case 56 2.5 Symmetry Properties and Conservation Laws 65 2.6 Conservation of Energy 71 2.7 Noether’s Theorem 73 Problems 78 3 Kinematics of Rotational Motion 86 3.1 Orthogonal Transformations 86 3.2 Possible Displacements of a Rigid Body 92 3.3 Euler Angles 95 3.4 Infinitesimal Rotations and Angular Velocity 96 3.5 Rotation Group and Infinitesimal Generators 102 3.6 Dynamics in Non-Inertial Reference Frames 103 Problems 109 4 Dynamics of Rigid Bodies 112 4.1 Angular Momentum and Inertia Tensor 112 4.2 Mathematical Interlude: Tensors and Dyadics 114 4.3 Moments and Products of Inertia 119 vii viii Contents 4.4 Kinetic Energy and Parallel Axis Theorem 120 4.5 Diagonalisation of the Inertia Tensor 122 4.6 Symmetries and Principal Axes of Inertia 126 4.7 Rolling Coin 129 4.8 Euler’s Equations and Free Rotation 131 4.9 Symmetric Top with One Point Fixed 139 Problems 145 5 Small Oscillations 149 5.1 One-Dimensional Case 149 5.2 Anomalous Case: Quartic Oscillator 154 5.3 Stationary Motion and Small Oscillations 161 5.4 Small Oscillations: General Case 163 5.5 Normal Modes of Vibration 165 5.6 Normal Coordinates 170 5.7 Mathematical Supplement 176 Problems 178 6 Relativistic Mechanics 183 6.1 Lorentz Transformations 183 6.2 Light Cone and Causality 188 6.3 Vectors and Tensors 191 6.4 Tensor Fields 194 6.5 Physical Laws in Covariant Form 196 6.6 Relativistic Dynamics 199 6.7 Relativistic Collisions 204 6.8 Relativistic Dynamics in Lagrangian Form 207 6.9 Action at a Distance in Special Relativity 209 Problems 211 7 Hamiltonian Dynamics 215 7.1 Hamilton’s Canonical Equations 215 7.2 Symmetries and Conservation Laws 220 7.3 The Virial Theorem 221 7.4 Relativistic Hamiltonian Formulation 224 7.5 Hamilton’s Equations in Variational Form 226 7.6 Time as a Canonical Variable 228 7.7 The Principle of Maupertuis 234 Problems 237 8 Canonical Transformations 242 8.1 Canonical Transformations and Generating Functions 242 8.2 Canonicity and Lagrange Brackets 248 ix Contents 8.3 Symplectic Notation 250 8.4 Poisson Brackets 254 8.5 Infinitesimal Canonical Transformations 258 8.6 Angular Momentum Poisson Brackets 262 8.7 Lie Series and Finite Canonical Transformations 264 8.8 Theorems of Liouville and Poincare´ 268 8.9 Constrained Hamiltonian Systems 272 Problems 281 9 The Hamilton-Jacobi Theory 288 9.1 The Hamilton-Jacobi Equation 288 9.2 One-Dimensional Examples 291 9.3 Separation of Variables 294 9.4 Incompleteness of the Theory: Point Charge in Dipole Field 299 9.5 Action as a Function of the Coordinates 303 9.6 Action-Angle Variables 306 9.7 Integrable Systems: The Liouville-Arnold Theorem 312 9.8 Non-integrability Criteria 316 9.9 Integrability, Chaos, Determinism and Ergodicity 318 9.10 Prelude to the KAM Theorem 320 9.11 Action Variables in the Kepler Problem 323 9.12 Adiabatic Invariants 325 9.13 Hamilton-Jacobi Theory and Quantum Mechanics 329 Problems 332 10 Hamiltonian Perturbation Theory 338 10.1 Statement of the Problem 338 10.2 Generating Function Method 341 10.3 One Degree of Freedom 342 10.4 Several Degrees of Freedom 345 10.5 The Kolmogorov-Arnold-Moser Theory 348 10.6 Stability: Eternal or Long Term 353 10.7 Lie Series Method 354 Problems 358 11 Classical Field Theory 360 11.1 Lagrangian Field Theory 360 11.2 Relativistic Field Theories 365 11.3 Functional Derivatives 367 11.4 Hamiltonian Field Theory 371 11.5 Symmetries of the Action and Noether’s Theorem 374 11.6 Solitary Waves and Solitons 378 11.7 Constrained Fields 381 Problems 385 x Contents Appendix A Indicial Notation 392 Appendix B Frobenius Integrability Condition 398 Appendix C Homogeneous Functions and Euler’s Theorem 406 Appendix D Vector Spaces and Linear Operators 408 Appendix E Stability of Dynamical Systems 421 Appendix F Exact Differentials 426 Appendix G Geometric Phases 428 Appendix H Poisson Manifolds 433 Appendix I Decay Rate of Fourier Coefficients 440 References 442 Index 452 Preface It is no exaggeration to say that analytical mechanics is the foundation of theoretical physics. The imposing edifice of quantum theory was erected on the basis of analytical mechanics, particularly in the form created by Hamilton. Statistical mechanics and the quantum field theory of elementary particles are strongly marked by structural elements extracted from classical mechanics. Moreover, the modern development of the theories of chaos and general dynamical systems has given rise to a renaissance of classical mechanics that began in the middle of the twentieth century and is ongoing today. Thus, the study of virtually any branch of contemporary physics requires a solid background in analytical mechanics, which remains important in itself on account of its applications in engineering and celestial mechanics. For more than two decades, intermittently, the author has taught analytical mechanics to undergraduate physics students at Universidade Federal Fluminense in Niteroi,´ Brazil. The present book, fruit of the author’s teaching experience and ceaseless fascination with the subject, aims at advanced undergraduate or graduate students who have previously taken a course in intermediate mechanics at the level of Marion & Thornton (1995) or Taylor (2005). As to mathematical prerequisites, the usual courses on calculus of one and several variables, ordinary differential equations and linear algebra are sufficient. Analytical mechanics is much more than a mere reformulation of Newtonian mechanics, and its methods have not been primarily conceived to help solve the equations of motion of specific mechanical systems. To make this clear, symmetry properties, invariance and general structural features of mechanics are highlighted in the course of the presentation. As remarked by Vladimir I. Arnold, numerous modern mathematical theories, with which some of the greatest names in the history of mathematics are associated, owe their existence to problems in mechanics.
Load more

Recommended publications
  • Covariant Hamiltonian Field Theory 3
    December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte COVARIANT HAMILTONIAN FIELD THEORY JURGEN¨ STRUCKMEIER and ANDREAS REDELBACH GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH Planckstr. 1, 64291 Darmstadt, Germany and Johann Wolfgang Goethe-Universit¨at Frankfurt am Main Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany [email protected] Received 18 July 2007 Revised 14 December 2020 A consistent, local coordinate formulation of covariant Hamiltonian field theory is pre- sented. Whereas the covariant canonical field equations are equivalent to the Euler- Lagrange field equations, the covariant canonical transformation theory offers more gen- eral means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinites- imal canonical transformation furnishes the most general form of Noether’s theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations. Keywords: Field theory; Hamiltonian density; covariant. PACS numbers: 11.10.Ef, 11.15Kc arXiv:0811.0508v6 [math-ph] 15 Dec 2020 1. Introduction Relativistic field theories and gauge theories are commonly formulated on the basis of a Lagrangian density L1,2,3,4. The space-time evolution of the fields is obtained by integrating the Euler-Lagrange field equations that follow from the four-dimensional representation of Hamilton’s action principle.
    [Show full text]
  • Description of Bound States and Scattering Processes Using Hamiltonian Renormalization Group Procedure for Effective Particles in Quantum field Theory
    Description of bound states and scattering processes using Hamiltonian renormalization group procedure for effective particles in quantum field theory Marek Wi˛eckowski June 2005 Doctoral thesis Advisor: Professor Stanisław D. Głazek Warsaw University Department of Physics Institute of Theoretical Physics Warsaw 2005 i Acknowledgments I am most grateful to my advisor, Professor Stanisław Głazek, for his invaluable advice and assistance during my work on this thesis. I would also like to thank Tomasz Masłowski for allowing me to use parts of our paper here. My heartfelt thanks go to Dr. Nick Ukiah, who supported and encouraged me during the writing of this thesis and who helped me in editing the final manuscript in English. Finally, I would like to thank my family, especially my grandmother, who has been a great support to me during the last few years. MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... ii MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... iii Abstract This thesis presents examples of a perturbative construction of Hamiltonians Hλ for effective particles in quantum field theory (QFT) on the light front. These Hamiltonians (1) have a well-defined (ultraviolet-finite) eigenvalue problem for bound states of relativistic constituent fermions, and (2) lead (in a scalar theory with asymptotic freedom in perturbation theory in third and partly fourth order) to an ultraviolet-finite and covariant scattering matrix, as the Feynman diagrams do. λ is a parameter of the renormalization group for Hamiltonians and qualitatively means the inverse of the size of the effective particles. The same procedure of calculating the operator Hλ applies in description of bound states and scattering.
    [Show full text]
  • Turbulence, Entropy and Dynamics
    TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions
    [Show full text]
  • Advances in Quantum Field Theory
    ADVANCES IN QUANTUM FIELD THEORY Edited by Sergey Ketov Advances in Quantum Field Theory Edited by Sergey Ketov Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Romana Vukelic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from [email protected] Advances in Quantum Field Theory, Edited by Sergey Ketov p.
    [Show full text]
  • Classical Hamiltonian Field Theory
    Classical Hamiltonian Field Theory Alex Nelson∗ Email: [email protected] August 26, 2009 Abstract This is a set of notes introducing Hamiltonian field theory, with focus on the scalar field. Contents 1 Lagrangian Field Theory1 2 Hamiltonian Field Theory2 1 Lagrangian Field Theory As with Hamiltonian mechanics, wherein one begins by taking the Legendre transform of the Lagrangian, in Hamiltonian field theory we \transform" the Lagrangian field treatment. So lets review the calculations in Lagrangian field theory. Consider the classical fields φa(t; x¯). We use the index a to indicate which field we are talking about. We should think of x¯ as another index, except it is continuous. We will use the confusing short hand notation φ for the column vector φ1; : : : ; φn. Consider the Lagrangian Z 3 L(φ) = L(φ, @µφ)d x¯ (1.1) all space where L is the Lagrangian density. Hamilton's principle of stationary action is still used to determine the equations of motion from the action Z S[φ] = L(φ, @µφ)dt (1.2) where we find the Euler-Lagrange equations of motion for the field d @L @L µ a = a (1.3) dx @(@µφ ) @φ where we note these are evil second order partial differential equations. We also note that we are using Einstein summation convention, so there is an implicit sum over µ but not over a. So that means there are n independent second order partial differential equations we need to solve. But how do we really know these are the correct equations? How do we really know these are the Euler-Lagrange equations for classical fields? We can obtain it directly from the action S by functional differentiation with respect to the field.
    [Show full text]
  • Higher Order Geometric Theory of Information and Heat Based On
    Article Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning Frédéric Barbaresco Department of Advanced Radar Concepts, Thales Land Air Systems, Voie Pierre-Gilles de Gennes, 91470 Limours, France; [email protected] Received: 9 August 2018; Accepted: 9 October; Published: 2 November 2018 Abstract: We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric.
    [Show full text]
  • Arxiv:1808.08689V3
    Hamiltonian Nature of Monopole Dynamics J. M. Heningera, P. J. Morrisona aDepartment of Physics and Institute for Fusion Studies, The University of Texas at Austin, Austin, TX, 78712, USA Abstract Classical electromagnetism with magnetic monopoles is not a Hamiltonian field theory because the Jacobi identity for the Poisson bracket fails. The Jacobi identity is recovered only if all of the species have the same ratio of electric to magnetic charge or if an electron and a monopole can never collide. Without the Jacobi identity, there are no local canonical coordinates or Lagrangian action principle. To build a quantum field of magnetic monopoles, we either must explain why the positions of electrons and monopoles can never coincide or we must resort to new quantization techniques. 1. Introduction This letter considers the classical gauge-free theory of electromagnetic fields interacting with electrically and magnetically charged matter as a Hamiltonian field theory. We begin with a brief history of magnetic monopoles and then describe why monopole theories are not Hamiltonian field theories. The modern theory of magnetic monopoles was developed by Dirac [1, 2]. He showed that an electron in the magnetic field of a monopole is equivalent to an electron whose wave function is zero along a semi-infinite ‘string’ extending from the location of the monopole. Along this string, the electromagnetic vector potential is undefined. The phase of the electron is no longer single valued along a loop encircling the Dirac string. In order for observables to be single valued, the phase shift must be an integer multiple of 2π, so the electric and magnetic charge must be quantized.
    [Show full text]
  • The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues
    Commun. Comput. Phys. Vol. 19, No. 3, pp. 758-769 doi: 10.4208/cicp.101114.140715a March 2016 The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues 1,2, 1,2 Christian Cherubini ∗ and Simonetta Filippi 1 Unit of Nonlinear Physics and Mathematical Modeling, Campus Bio-Medico, University of Rome, I-00128, Rome, Italy. 2 International Center for Relativistic Astrophysics – I.C.R.A., Campus Bio-Medico, University of Rome, I-00128, Rome, Italy. Communicated by Lianjie Huang Received 10 November 2014; Accepted (in revised version) 14 July 2015 Abstract. The Von Mises quasi-linear second order wave equation, which completely describes an irrotational, compressible and barotropic classical perfect fluid, can be de- rived from a nontrivial least action principle for the velocity scalar potential only, in contrast to existing analog formulations which are expressed in terms of coupled den- sity and velocity fields. In this article, the classical Hamiltonian field theory specifically associated to such an equation is developed in the polytropic case and numerically verified in a simplified situation. The existence of such a mathematical structure sug- gests new theoretical schemes possibly useful for performing numerical integrations of fluid dynamical equations. Moreover it justifies possible new functional forms for La- grangian densities and associated Hamiltonian functions in other theoretical classical physics contexts. AMS subject classifications: 76A02, 65M60, 37K05 PACS: 47.40.-x, 02.30.Jr, 11.10.Ef, 04.90.+e Key words: Classical field theories, fluid dynamics, partial differential equations, numerical methods. 1 Introduction The dynamics of non viscous barotropic fluids is characterized by well defined Lagrangian [1] as well as Hamiltonian classical field theory formulations [1–6] developed in terms of coupled density and velocity fields.
    [Show full text]
  • The Lagrangian-Hamiltonian Formalism for Higher Order Field Theories
    The Di©ety Institute Preprint Series Preprint DIPS2/2009 May 28, 2009 The Lagrangian-Hamiltonian Formalism for Higher Order Field Theories by L. Vitagliano Available via INTERNET: http://di©ety.ac.ru; http://di©ety.org The Di©ety Institute DIPS-2/2009 arXiv:0905.4580 The Lagrangian-Hamiltonian Formalism for Higher Order Field Theories L. VITAGLIANO Abstract. We generalize the lagrangian-hamiltonian formalism of Skinner and Rusk to higher order ¦eld theories on ¦ber bundles. As a byproduct we solve the long standing problem of de¦ning, in a coordinate free manner, a hamiltonian formalism for higher order lagrangian ¦eld theories, which does only depend on the action func- tional and, therefore, unlike previously proposed formalisms, is free from any relevant ambiguity. Introduction First order lagrangian mechanics can be generalized to higher order lagrangian ¦eld theory. Moreover, the latter has got a very elegant geometric (and homological) for- mulation (see, for instance, [1]) on which there is general consensus. On the other side, it seems that the generalization of hamiltonian mechanics of lagrangian systems to higher order ¦eld theory presents some more problems. They have been proposed several answers (see, for instance, [2, 3, 4, 5, 6, 7, 8, 9, 10] and references therein) to the question: is there any reasonable, higher order, ¦eld theoretic analogue of hamiltonian mechanics? To our opinion, non of them is satisfactorily natural, especially because of the common emergence of ambiguities due to either the arbitrary choice of a coordinate system [2] or the choice of a Legendre transform [7, 8, 10]. Namely, the latter seems not to be uniquely de¦nable, except in the case of ¦rst order lagrangian ¦eld theories when a satisfactory hamiltonian formulation can be presented in terms of multisym- plectic geometry (see, for instance, [11] - see also [12] for a recent review, and references therein).
    [Show full text]
  • Introduction to Relativistic Quantum Field Theory
    Introduction to Relativistic Quantum Field Theory Prof. Dr. Stefan Dittmaier, Dr. Heidi Rzehak and Dr. Christian Schwinn Albert-Ludwigs-Universit¨at Freiburg, Physikalisches Institut D-79104 Freiburg, Germany Summer-Semester 2014 Draft: October 1, 2014 2 Contents 1 Introduction 7 I Quantization of Scalar Fields 9 2 Recapitulation of Special Relativity 11 2.1 Lorentz transformations, four-vectors, tensors . ......... 11 2.1.1 Minkowskispace ............................ 11 2.1.2 Lorentztransformations . 12 2.1.3 Differentialoperators.......................... 14 2.1.4 Tensors ................................. 15 2.2 LorentzgroupandLorentzalgebra . 15 2.2.1 Classification of Lorentz transformations . 15 2.2.2 Infinitesimal transformations and group generators . ... 17 2.3 PoincaregroupandPoincarealgebra . 19 2.3.1 Thebasics................................ 19 2.3.2 Generatorsasdifferentialoperators . 20 2.4 Relativistic point particles . 22 3 The Klein–Gordon equation 25 3.1 Relativistic wave equation . 25 3.2 Solutions of the Klein–Gordon equation . 26 3.3 Conservedcurrent................................ 28 3.4 Interpretation of the Klein–Gordon equation . ... 29 4 Classical Field Theory 31 4.1 Lagrangian and Hamiltonian formalism . 31 4.1.1 Lagrangianfieldtheory......................... 31 4.1.2 Hamiltonian field theory . 35 4.2 Actionsforscalarfields............................. 36 4.2.1 Freerealscalarfield .......................... 36 4.2.2 Freecomplexscalarfield ........................ 37 4.3 Interactingfields ................................ 38 3 4 CONTENTS 4.3.1 Scalar self-interactions . 38 4.3.2 Explicit calculation of the Green function (propagator) ....... 40 4.4 SymmetriesandtheNoetherTheorem . 43 4.4.1 Continuoussymmetries. .. .. 43 4.4.2 DerivationoftheNoethertheorem . 45 4.4.3 Internal symmetries and conserved currents . .. 46 4.4.4 Translation invariance and energy-momentum tensor . 47 5 Canonical quantization of free scalar fields 49 5.1 Canonical commutation relations .
    [Show full text]
  • Hamiltonian Formulation of Two-Dimensional Gyroviscous MHD Philip 1
    Hamiltonian Formulation of Two-Dimensional Gyroviscous MHD Philip 1. Morrison *, I. L. Caldas **, and H. Tasso Max-Planck-Instilut fUr Plasma physik, EURATOM Assnciation, Garching Z. Naturforsch. 39., 1023-1027 (1984); received August 9,1984 Gyroviscous MHD in two dimensions is shown to be a Hamiltonian field theory in terms of a rton~canonical Poisson bracket. This bracket is of the Lie-Poisson type, but possesses an unfamiliar inner Lie algebra. Analysis of this algebra motivates a transformation that enables a Clebsch-like potential decomposition that makes Lagrangian and canonical Hamiltonian formulations possible. I. Introduction and then present the Poisson bracket for the gyro­ This work is concerned with the Hamiltonian MHD model. Section IV is concerned with the algebraic interprctation of the bracket presented in field formulation of the equations that describe Sect. III. In case the methods of this section are two-dimensional non-dissipative gyroviscous one­ unfamiliar to the reader, it may be skipped without fluid plasmas [I, 2, 3, 4J. These equations differ loss of continuity. Section V deals with Clebsch [13J from the usual Eulerian form of ideal MHO by the . potentials, i. e. a potential decomposition that trans­ inclusion of a non-entropy-producing stress tensor that arises from the finiteness of the ion gyroradius. forms the noncanonical formalism into canonical The physics of this stress \ensor is important for form with gauge conditions. We thus obtain a varia­ tional principle. Clebsch-potentials are discussed in modelling tokamak discharges [5J, and may be important for calculation of the MHO k-spectrum [9], [12J, [14J-[18J.
    [Show full text]
  • Hamiltonian Field Theory
    Hamiltonian Field Theory August 31, 2016 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 and is best suited for quantizing a field theory using the path- integral (aka functional integral) approach. However, a quicker route to quantizing a classical field theory is thru the \canonical quantization" program which is what you will find in most introductory treatments on quantum field theory. So for us, the Lagrangian is just a crutch to extract the Hamiltonian. However there is a price to pay when we switch from a Lagrangian approach to a Hamiltonian approach, that is manifest Lorentz invariance. Recall that a field theory is a system where we have a degree of freedom located at each point in space. If this degree of freedom is a Lorentz scalar, we will denote the degree of freedom located at position x by Φ(x) and its conjugate momenta, Π(x). These degrees of freedom of course fluctuate in time, so when we include this time-dependence, the notation becomes, Φ(x; t) and Π(x; t): The job of Hamiltonian field theory is describe time-evolution of Π and Φ in equations as follows, Π_ = :::; Φ_ = ::: The dot is time derivative, i.e. we have to choose a time axis or equivalently preferred reference frame which breaks manifest Lorentz invariance. Recall that earlier, in the Lagrangian/action approach we did not have to choose a time-frame because the equation of motion looked Lorentz invariant, e.g.
    [Show full text]