Hamilton s principle and Symmetries
|
|
- Brandon Simmons
- 5 years ago
- Views:
Transcription
1 Hamilton s principle and Symmetries Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2011 August 18, 2011
2 The Hamiltonian The change in the Lagrangian due to a virtual change of coordinates is dl = k ṗ k dq k +p k d q k. Using this, one can define a function, called the Hamiltonian, H(p k,q k ), by eliminating q k to write [ ] dh = d p k q k L = q k dp k ṗ k dq k. k k Since p k and q k are independent variables, one has Hamilton s equations q k = H and ṗ k = H. p k q k
3 An example For a single particle moving in a potential, one can write L = 1 2 m ẋ 2 V(x), and p = L ẋ = mẋ. Eliminating ẋ, the Hamiltonian is Hamilton s equations are H = p ẋ L = 1 2m p 2 +V(x). ẋ = 1 p, and ṗ = V(x). m These are exactly the usual equations of motion.
4 Poisson brackets Take any two functions on phase space, f(q k,p k ) and g(q k,p k ). Then we define the Poisson bracket of these two functions as [f,g] = [ f g g ] f. q k p k q k p k k Clearly, interchanging f and g in the expression on the right changes the sign. So [f,g] = [g,f]. Also, because of this antisymmetry, [f,f] = 0. Clearly, the time derivative of f is given by the expression df dt = k = k [ f q k + f ] ṗ k q k p k [ f H f H q k p k p k q k ] = [f,h]. We have used the EoM to get the second line from the first.
5 Time independent Hamiltonians are conserved More generally, for any time-dependent function on phase space f(q k,p k,t), one has df dt = f t +[f,h]. It follows trivially that, for the Hamiltonian itself we have the identity dh dt = H t, so that if the Hamiltonian is time independent, then it is conserved. Other identities for the Poisson bracket include the Jacobi identity [[f,g],h]+[[g,h],f]+[[h,f],g] = 0. Problem 11 Prove the Jacobi identity.
6 Elementary Poisson brackets Since all the q k and p k are independent variables, their derivatives with respect to each other vanish. So we have [q j,q k ] = 0 and [p j,p k ] = 0. Also, [q i,p j ] = [ qi p j p ] j q i = δ ik δ jk = δ ij. q k p k q k p k k k The E0M can be written in the form q i = [q i,h] and ṗ i = [p i,h]. Using the definition of the Poisson bracket, one sees that these reproduce the canonical equations. If some momenta are conserved, then one clearly has H/ q i = 0. The Hamiltonian does not depend on the corresponding coordinates, i.e., we have a symmetry.
7 Angular momenta rotate vectors We write the components of the angular momentum L = x p using the Levi-Civita tensor as L i = ǫ ijk x j p k. In order to evaluate the Poisson bracket [L i,x j ] we note that for any three functions on phase space, f, g, h, Using this we find [fg,h] = f[g,h]+g[f,h]. [L i,x j ] = ǫ iαβ [x α p β,x j ] = ǫ iαβ {x α [p β,x j ]+p β [x α,x j ]} = ǫ iαβ x α δ βj = ǫ ijk x k. This is the definition of a vector function on phase space. Problem 12 Prove that the momentum p is a vector function on phase space.
8 Poisson brackets of angular momenta The Poisson bracket of two components can be easily worked out [L i,l j ] = ǫ iαβ ǫ jγδ [x α p β,x γ p δ ]. Using the identity for Poisson brackets of products, we can write [L i,l j ] = ǫ iαβ ǫ jγδ {x α [p β,x γ p δ ]+p β [x α,x γ p δ ]}. Using the identity a second time we can write [L i,l j ] = ǫ iαβ ǫ jγδ {x α x γ [p β,p δ ]+x α p δ [p β,x γ ] +p β x γ [x α,p δ ]+p β p δ [x α,x γ ]} = ǫ iαβ ǫ jγδ { x α p δ δ βγ +p β x γ δ αδ }
9 The angular momentum is a vector We have the tensor identity ǫ iαβ ǫ jγβ = δ ij δ αγ δ iγ δ jα, where we have used the summation convention repeated indices are summed. Using this we reduce the Poisson bracket [L i,l j ] = ǫ iαβ ǫ jγδ { x α p δ δ βγ +p β x γ δ αδ } = ǫ iαβ ǫ jγβ {x α p γ x γ p α } renaming dummy indices = {δ ij δ αγ δ iγ δ jα } {x α p γ x γ p α } = x i p j x j p i = ǫ ijk L k. This completes the demonstration that the angular momentum itself is a vector.
10 Keywords and References Keywords Hamiltonian, Hamilton s equations, Poisson bracket, antisymmetry, Jacobi identity, conserved, symmetry, Levi-Civita tensor, vector function References Goldstein, Chapters 8, 9 Landau, Sections 40, 42
Homework 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 informationGeneralized Coordinates, Lagrangians
Generalized Coordinates, Lagrangians Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 August 10, 2012 Generalized coordinates Consider again the motion of a simple pendulum. Since it is one
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 informationHANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS
MATHEMATICS 7302 (Analytical Dynamics) YEAR 2016 2017, TERM 2 HANDOUT #12: THE HAMILTONIAN APPROACH TO MECHANICS These notes are intended to be read as a supplement to the handout from Gregory, Classical
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 information(Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7) p j . (5.1) !q j. " d dt = 0 (5.2) !p j . (5.
Chapter 5. Hamiltonian Dynamics (Most of the material presented in this chapter is taken from Thornton and Marion, Chap. 7) 5.1 The Canonical Equations of Motion As we saw in section 4.7.4, the generalized
More informationNIU PHYS 500, Fall 2006 Classical Mechanics Solutions for HW6. Solutions
NIU PHYS 500, Fall 006 Classical Mechanics Solutions for HW6 Assignment: HW6 [40 points] Assigned: 006/11/10 Due: 006/11/17 Solutions P6.1 [4 + 3 + 3 = 10 points] Consider a particle of mass m moving in
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 informationCoordinates, phase space, constraints
Coordinates, phase space, constraints Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2011 August 1, 2011 Mechanics of a single particle For the motion of a particle (of constant mass m and position
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 informationPHY 5246: Theoretical Dynamics, Fall November 16 th, 2015 Assignment # 11, Solutions. p θ = L θ = mr2 θ, p φ = L θ = mr2 sin 2 θ φ.
PHY 5246: Theoretical Dynamics, Fall 215 November 16 th, 215 Assignment # 11, Solutions 1 Graded problems Problem 1 1.a) The Lagrangian is L = 1 2 m(ṙ2 +r 2 θ2 +r 2 sin 2 θ φ 2 ) V(r), (1) and the conjugate
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 informationm 1 m 2 π/2 θ NATURAL SCIENCES TRIPOS Wednesday 16 January 2013 THEORETICAL PHYSICS I Answers
NTURL SCIENCES TRIPOS Part II Wednesday 16 January 013 THEORETICL PHYSICS I nswers 1 Consider a double pendulum composed of two masses m 1 and m attached to two rigid massless rods of equal length l as
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 informationEnergy and Equations of Motion
Energy and Equations of Motion V. Tanrıverdi tanriverdivedat@googlemail.com Physics Department, Middle East Technical University, Ankara / TURKEY Abstract. From the total time derivative of energy an equation
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 informationMATHEMATICAL PHYSICS
MATHEMATICAL PHYSICS Third Year SEMESTER 1 015 016 Classical Mechanics MP350 Prof. S. J. Hands, Prof. D. M. Heffernan, Dr. J.-I. Skullerud and Dr. M. Fremling Time allowed: 1 1 hours Answer two questions
More informationCanonical transformations (Lecture 4)
Canonical transformations (Lecture 4) January 26, 2016 61/441 Lecture outline We will introduce and discuss canonical transformations that conserve the Hamiltonian structure of equations of motion. Poisson
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 information1 Hamiltonian formalism
1 Hamiltonian formalism 1.1 Hamilton s principle of stationary action A dynamical system with a finite number n degrees of freedom can be described by real functions of time q i (t) (i =1, 2,..., n) which,
More informationCoordinates, phase space, constraints
Coordinates, phase space, constraints Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 6 August, 2012 Mechanics of a single particle For the motion of a particle (of constant mass m and position
More informationHamilton-Jacobi Formulation of A Non-Abelian Yang-Mills Theories
EJTP 5, No. 17 (2008) 65 72 Electronic Journal of Theoretical Physics Hamilton-Jacobi Formulation of A Non-Abelian Yang-Mills Theories W. I. Eshraim and N. I. Farahat Department of Physics Islamic University
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 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 informationHamiltonian Solution I
Physics 4 Lecture 5 Hamiltonian Solution I Lecture 5 Physics 4 Classical Mechanics II September 7th 2007 Here we continue with the Hamiltonian formulation of the central body problem we will uncover the
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 informationarxiv:physics/ v1 [physics.class-ph] 3 Apr 2000
Dirac monopole with Feynman brackets Alain Bérard arxiv:physics/0004008v1 [physicsclass-ph] 3 Apr 2000 LPLI-Institut de Physique, 1 blvd DArago, F-57070 Metz, France Y Grandati LPLI-Institut de Physique,
More informationAn Exactly Solvable 3 Body Problem
An Exactly Solvable 3 Body Problem The most famous n-body problem is one where particles interact by an inverse square-law force. However, there is a class of exactly solvable n-body problems in which
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 informationPhysics 342 Lecture 26. Angular Momentum. Lecture 26. Physics 342 Quantum Mechanics I
Physics 342 Lecture 26 Angular Momentum Lecture 26 Physics 342 Quantum Mechanics I Friday, April 2nd, 2010 We know how to obtain the energy of Hydrogen using the Hamiltonian operator but given a particular
More informationAdding angular momenta
Adding angular momenta Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 Eleventh Lecture Outline 1 Outline 2 Some definitions 3 The simplest example: summing two momenta 4 Interesting physics: summing
More informationPhysics 339 Euler Angles & Free Precession November 2017 Hamilton s Revenge
Physics 339 Euler Angles & Free Precession November 217 Hamilton s Revenge As described in the textbook, Euler Angles are a way to specify the configuration of a 3d object. Starting from a fixed configuration
More informationOn complexified quantum mechanics and space-time
On complexified quantum mechanics and space-time Dorje C. Brody Mathematical Sciences Brunel University, Uxbridge UB8 3PH dorje.brody@brunel.ac.uk Quantum Physics with Non-Hermitian Operators Dresden:
More informationCourse 241: Advanced Mechanics. Scholarship Questions
Course 41: Advanced echanics Scholarship Questions This covers the first sixteen questions from part I up to the end of rigid bodies, and a selection of the questions from part II. I wouldn t be too bothered
More informationNewton s Second Law in a Noncommutative Space
Newton s Second Law in a Noncommutative Space Juan M. Romero, J.A. Santiago and J. David Vergara Instituto de Ciencias Nucleares, U.N.A.M., Apdo. Postal 70-543, México D.F., México sanpedro, santiago,
More informationPhysics of Continuous media
Physics of Continuous media Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 October 26, 2012 Deformations of continuous media If a body is deformed, we say that the point which originally had
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 informationQuantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams
Quantum Field Theory I Examination questions will be composed from those below and from questions in the textbook and previous exams III. Quantization of constrained systems and Maxwell s theory 1. The
More informationNon-Commutative Worlds
Non-Commutative Worlds Louis H. Kauffman*(kauffman@uic.edu) We explore how a discrete viewpoint about physics is related to non-commutativity, gauge theory and differential geometry.
More informationHAMILTON S PRINCIPLE
HAMILTON S PRINCIPLE In our previous derivation of Lagrange s equations we started from the Newtonian vector equations of motion and via D Alembert s Principle changed coordinates to generalised coordinates
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 informationMassachusetts Institute of Technology Department of Physics. Final Examination December 17, 2004
Massachusetts Institute of Technology Department of Physics Course: 8.09 Classical Mechanics Term: Fall 004 Final Examination December 17, 004 Instructions Do not start until you are told to do so. Solve
More informationQuantization of scalar fields
Quantization of scalar fields March 8, 06 We have introduced several distinct types of fields, with actions that give their field equations. These include scalar fields, S α ϕ α ϕ m ϕ d 4 x and complex
More informationChapter 2 Classical Mechanics
Chapter 2 Classical Mechanics Une intelligence qui, à un instant donné, connaîtrait toutes les forces dont la nature est animée et la situation respective des êtres qui la compose embrasserait dans la
More informationGravity theory on Poisson manifold with R-flux
Gravity theory on Poisson manifold with R-flux Hisayoshi MURAKI (University of Tsukuba) in collaboration with Tsuguhiko ASAKAWA (Maebashi Institute of Technology) Satoshi WATAMURA (Tohoku University) References
More informationEuclidean path integral formalism: from quantum mechanics to quantum field theory
: from quantum mechanics to quantum field theory Dr. Philippe de Forcrand Tutor: Dr. Marco Panero ETH Zürich 30th March, 2009 Introduction Real time Euclidean time Vacuum s expectation values Euclidean
More informationEULER-LAGRANGE TO HAMILTON. The goal of these notes is to give one way of getting from the Euler-Lagrange equations to Hamilton s equations.
EULER-LAGRANGE TO HAMILTON LANCE D. DRAGER The goal of these notes is to give one way of getting from the Euler-Lagrange equations to Hamilton s equations. 1. Euler-Lagrange to Hamilton We will often write
More informationCanonical Quantization
Canonical Quantization March 6, 06 Canonical quantization of a particle. The Heisenberg picture One of the most direct ways to quantize a classical system is the method of canonical quantization introduced
More informationCE 530 Molecular Simulation
CE 530 Molecular Simulation Lecture Molecular Dynamics Simulation David A. Kofke Department of Chemical Engineering SUNY Buffalo kofke@eng.buffalo.edu MD of hard disks intuitive Review and Preview collision
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 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 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 informationExamples of the Dirac approach to dynamics of systems with constraints
Physica A 290 (2001) 431 444 www.elsevier.com/locate/physa Examples of the Dirac approach to dynamics of systems with constraints Sonnet Nguyen, Lukasz A. Turski Center for Theoretical Physics, College
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 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 informationAppendix to Lecture 2
PHYS 652: Astrophysics 1 Appendix to Lecture 2 An Alternative Lagrangian In class we used an alternative Lagrangian L = g γδ ẋ γ ẋ δ, instead of the traditional L = g γδ ẋ γ ẋ δ. Here is the justification
More informationPath Integral Quantization of the Electromagnetic Field Coupled to A Spinor
EJTP 6, No. 22 (2009) 189 196 Electronic Journal of Theoretical Physics Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor Walaa. I. Eshraim and Nasser. I. Farahat Department of
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 informationPhysical Dynamics (PHY-304)
Physical Dynamics (PHY-304) Gabriele Travaglini March 31, 2012 1 Review of Newtonian Mechanics 1.1 One particle Lectures 1-2. Frame, velocity, acceleration, number of degrees of freedom, generalised coordinates.
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationarxiv:gr-qc/ v2 6 Apr 1999
1 Notations I am using the same notations as in [3] and [2]. 2 Temporal gauge - various approaches arxiv:gr-qc/9801081v2 6 Apr 1999 Obviously the temporal gauge q i = a i = const or in QED: A 0 = a R (1)
More informationHamilton-Jacobi approach to Berezinian singular systems
1 Hamilton-Jacobi approach to erezinian singular systems M Pimentel and R G Teixeira Instituto de Física Teórica Universidade Estadual Paulista Rua Pamplona 145 01405-900 - São Paulo, SP razil and J L
More informationSecond quantization: where quantization and particles come from?
110 Phys460.nb 7 Second quantization: where quantization and particles come from? 7.1. Lagrangian mechanics and canonical quantization Q: How do we quantize a general system? 7.1.1.Lagrangian Lagrangian
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 informationThe kinetic equation (Lecture 11)
The kinetic equation (Lecture 11) January 29, 2016 190/441 Lecture outline In the preceding lectures we focused our attention on a single particle motion. In this lecture, we will introduce formalism for
More information2 Classical Field Theory
2 Classical Field Theory In what follows we will consider rather general field theories. The only guiding principles that we will use in constructing these theories are a) symmetries and b) a generalized
More informationTailoring BKL to Loop Quantum Cosmology
Tailoring BKL to Loop Quantum Cosmology Adam D. Henderson Institute for Gravitation and the Cosmos Pennsylvania State University LQC Workshop October 23-25 2008 Introduction Loop Quantum Cosmology: Big
More informationEquivalence of superintegrable systems in two dimensions
Equivalence of superintegrable systems in two dimensions J. M. Kress 1, 1 School of Mathematics, The University of New South Wales, Sydney 058, Australia. In two dimensions, all nondegenerate superintegrable
More informationRepresentations of angular momentum
Representations of angular momentum Sourendu Gupta TIFR Graduate School Quantum Mechanics 1 September 26, 2008 Sourendu Gupta (TIFR Graduate School) Representations of angular momentum QM I 1 / 15 Outline
More informationUnimodularity and preservation of measures in nonholonomic mechanics
Unimodularity and preservation of measures in nonholonomic mechanics Luis García-Naranjo (joint with Y. Fedorov and J.C. Marrero) Mathematics Department ITAM, Mexico City, MEXICO ẋ = f (x), x M n, f smooth
More informationFrom quantum to classical statistical mechanics. Polyatomic ideal gas.
From quantum to classical statistical mechanics. Polyatomic ideal gas. Peter Košovan peter.kosovan@natur.cuni.cz Dept. of Physical and Macromolecular Chemistry Lecture 5, Statistical Thermodynamics, MC260P105,
More informationVariable-separation theory for the null Hamilton Jacobi equation
JOURNAL OF MATHEMATICAL PHYSICS 46, 042901 2005 Variable-separation theory for the null Hamilton Jacobi equation S. Benenti, C. Chanu, and G. Rastelli Dipartimento di Matematica, Università di Torino,
More informationPhysics 5153 Classical Mechanics. The Virial Theorem and The Poisson Bracket-1
Physics 5153 Classical Mechanics The Virial Theorem an The Poisson Bracket 1 Introuction In this lecture we will consier two applications of the Hamiltonian. The first, the Virial Theorem, applies to systems
More informationOn singular lagrangians and Dirac s method
INVESTIGACIÓN Revista Mexicana de Física 58 (01 61 68 FEBRERO 01 On singular lagrangians and Dirac s method J.U. Cisneros-Parra Facultad de Ciencias, Universidad Autonoma de San Luis Potosi, Zona Uniiversitaria,
More informationThe Dirac Field. Physics , Quantum Field Theory. October Michael Dine Department of Physics University of California, Santa Cruz
Michael Dine Department of Physics University of California, Santa Cruz October 2013 Lorentz Transformation Properties of the Dirac Field First, rotations. In ordinary quantum mechanics, ψ σ i ψ (1) is
More informationAssignment 8. [η j, η k ] = J jk
Assignment 8 Goldstein 9.8 Prove directly that the transformation is canonical and find a generating function. Q 1 = q 1, P 1 = p 1 p Q = p, P = q 1 q We can establish that the transformation is canonical
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 informationLecture 4. Alexey Boyarsky. October 6, 2015
Lecture 4 Alexey Boyarsky October 6, 2015 1 Conservation laws and symmetries 1.1 Ignorable Coordinates During the motion of a mechanical system, the 2s quantities q i and q i, (i = 1, 2,..., s) which specify
More informationEOS 352 Continuum Dynamics Conservation of angular momentum
EOS 352 Continuum Dynamics Conservation of angular momentum c Christian Schoof. Not to be copied, used, or revised without explicit written permission from the copyright owner The copyright owner explicitly
More informationThe Physical Pendulum
The Physical Pendulum Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2011 August 20, 2011 The physical pendulum θ L T mg We choose to work with a generalized coordinate q which is the angle of
More informationDynamics of Relativistic Particles and EM Fields
October 7, 2008 1 1 J.D.Jackson, Classical Electrodynamics, 3rd Edition, Chapter 12 Lagrangian Hamiltonian for a Relativistic Charged Particle The equations of motion [ d p dt = e E + u ] c B de dt = e
More informationThe Matrix Representation of a Three-Dimensional Rotation Revisited
Physics 116A Winter 2010 The Matrix Representation of a Three-Dimensional Rotation Revisited In a handout entitled The Matrix Representation of a Three-Dimensional Rotation, I provided a derivation of
More informationarxiv:math-ph/ v1 27 Feb 2007
Goldfish geodesics and Hamiltonian reduction of matrix dynamics arxiv:math-ph/07009v 7 Feb 007 Joakim Arnlind, Martin Bordemann, Jens Hoppe, Choonkyu Lee Department of Mathematics, Royal Institute of Technology,
More informationSolutions to Problems in Goldstein, Classical Mechanics, Second Edition. Chapter 9
Solutions to Problems in Goldstein, Classical Mechanics, Second Edition Homer Reid October 29, 2002 Chater 9 Problem 9. One of the attemts at combining the two sets of Hamilton s equations into one tries
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 informationThe Particle-Field Hamiltonian
The Particle-Field Hamiltonian For a fundamental understanding of the interaction of a particle with the electromagnetic field we need to know the total energy of the system consisting of particle and
More information(a p (t)e i p x +a (t)e ip x p
5/29/3 Lecture outline Reading: Zwiebach chapters and. Last time: quantize KG field, φ(t, x) = (a (t)e i x +a (t)e ip x V ). 2Ep H = ( ȧ ȧ(t)+ 2E 2 E pa a) = p > E p a a. P = a a. [a p,a k ] = δ p,k, [a
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 informationQuantum Theory and Group Representations
Quantum Theory and Group Representations Peter Woit Columbia University LaGuardia Community College, November 1, 2017 Queensborough Community College, November 15, 2017 Peter Woit (Columbia University)
More informationClassical Mechanics and Statistical/Thermodynamics. January 2015
Classical Mechanics and Statistical/Thermodynamics January 2015 1 Handy Integrals: Possibly Useful Information 0 x n e αx dx = n! α n+1 π α 0 0 e αx2 dx = 1 2 x e αx2 dx = 1 2α 0 x 2 e αx2 dx = 1 4 π α
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 informationM. van Berkel DCT
Explicit solution of the ODEs describing the 3 DOF Control Moment Gyroscope M. van Berkel DCT 28.14 Traineeship report Coach(es): Supervisor: dr. N. Sakamoto Prof. dr. H. Nijmeijer Technische Universiteit
More informationLecture II: Hamiltonian formulation of general relativity
Lecture II: Hamiltonian formulation of general relativity (Courses in canonical gravity) Yaser Tavakoli December 16, 2014 1 Space-time foliation The Hamiltonian formulation of ordinary mechanics is given
More information-state problems and an application to the free particle
-state problems and an application to the free particle Sourendu Gupta TIFR, Mumbai, India Quantum Mechanics 1 2013 3 September, 2013 Outline 1 Outline 2 The Hilbert space 3 A free particle 4 Keywords
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 informationPerelman s Dilaton. Annibale Magni (TU-Dortmund) April 26, joint work with M. Caldarelli, G. Catino, Z. Djadly and C.
Annibale Magni (TU-Dortmund) April 26, 2010 joint work with M. Caldarelli, G. Catino, Z. Djadly and C. Mantegazza Topics. Topics. Fundamentals of the Ricci flow. Topics. Fundamentals of the Ricci flow.
More informationQuantization of Scalar Field
Quantization of Scalar Field Wei Wang 2017.10.12 Wei Wang(SJTU) Lectures on QFT 2017.10.12 1 / 41 Contents 1 From classical theory to quantum theory 2 Quantization of real scalar field 3 Quantization of
More informationarxiv:hep-th/ v1 23 Mar 1995
Straight-line string with curvature L.D.Soloviev Institute for High Energy Physics, 142284, Protvino, Moscow region, Russia arxiv:hep-th/9503156v1 23 Mar 1995 Abstract Classical and quantum solutions for
More informationLecture 19 (Nov. 15, 2017)
Lecture 19 8.31 Quantum Theory I, Fall 017 8 Lecture 19 Nov. 15, 017) 19.1 Rotations Recall that rotations are transformations of the form x i R ij x j using Einstein summation notation), where R is an
More informationVECTORS, TENSORS AND INDEX NOTATION
VECTORS, TENSORS AND INDEX NOTATION Enrico Nobile Dipartimento di Ingegneria e Architettura Università degli Studi di Trieste, 34127 TRIESTE March 5, 2018 Vectors & Tensors, E. Nobile March 5, 2018 1 /
More information