Classical Mechanics - Baez

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    Lectures on

    Classical Mechanics

    by

    John C. Baez

    notes by Derek K. Wise

    Department of Mathematics

    University of California, Riverside

    LaTeXed by Blair Smith

    Department of Physics and Astronomy

    Louisiana State University

    2005

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    c2005 John C. Baez & Derek K. Wise

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    Preface

    These are notes for a mathematics graduate course on classical mechanics at U.C.Riverside. Ive taught this course three times recently. Twice I focused on the Hamiltonianapproach. In 2005 I started with the Lagrangian approach, with a heavy emphasis onaction principles, and derived the Hamiltonian approach from that. This approach seemsmore coherent.

    Derek Wise took beautiful handwritten notes on the 2005 course, which can be foundon my website:

    http://math.ucr.edu/home/baez/classical/

    Later, Blair Smith from Louisiana State University miraculously appeared and volun-teered to turn the notes into LATEX . While not yet the book Id eventually like to write,the result may already be helpful for people interested in the mathematics of classicalmechanics.

    The chapters in this LATEX version are in the same order as the weekly lectures, butIve merged weeks together, and sometimes split them over chapter, to obtain a moretextbook feel to these notes. For reference, the weekly lectures are outlined here.

    Week 1: (Mar. 28, 30, Apr. 1)The Lagrangian approach to classical mechanics:deriving F = ma from the requirement that the particles path be a critical point of

    the action. The prehistory of the Lagrangian approach: DAlemberts principle of leastenergy in statics, Fermats principle of least time in optics, and how DAlembertgeneralized his principle from statics to dynamics using the concept of inertia force.

    Week 2: (Apr. 4, 6, 8)Deriving the Euler-Lagrange equations for a particle on anarbitrary manifold. Generalized momentum and force. Noethers theorem on conservedquantities coming from symmetries. Examples of conserved quantities: energy, momen-tum and angular momentum.

    Week 3 (Apr. 11, 13, 15)Example problems: (1) The Atwood machine. (2) Africtionless mass on a table attached to a string threaded through a hole in the table, witha mass hanging on the string. (3) A special-relativistic free particle: two Lagrangians, one

    with reparametrization invariance as a gauge symmetry. (4) A special-relativistic chargedparticle in an electromagnetic field.Week 4 (Apr. 18, 20, 22)More example problems: (4) A special-relativistic charged

    particle in an electromagnetic field in special relativity, continued. (5) A general-relativisticfree particle.

    Week 5 (Apr. 25, 27, 29)How Jacobi unified Fermats principle of least time andLagranges principle of least action by seeing the classical mechanics of a particle in apotential as a special case of optics with a position-dependent index of refraction. Theubiquity of geodesic motion. Kaluza-Klein theory. From Lagrangians to Hamiltonians.

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    Week 6 (May 2, 4, 6)From Lagrangians to Hamiltonians, continued. Regular and

    strongly regular Lagrangians. The cotangent bundle as phase space. Hamiltons equa-tions. Getting Hamiltons equations directly from a least action principle.

    Week 7 (May 9, 11, 13)Waves versus particles: the Hamilton-Jacobi equation.Hamiltons principal function and extended phase space. How the Hamilton-Jacobi equa-tion foreshadows quantum mechanics.

    Week 8 (May 16, 18, 20)Towards symplectic geometry. The canonical 1-form andthe symplectic 2-form on the cotangent bundle. Hamiltons equations on a symplecticmanifold. Darbouxs theorem.

    Week 9 (May 23, 25, 27)Poisson brackets. The Schrodinger picture versus theHeisenberg picture in classical mechanics. The Hamiltonian version of Noethers theorem.Poisson algebras and Poisson manifolds. A Poisson manifold that is not symplectic.Liouvilles theorem. Weils formula.

    Week 10 (June 1, 3, 5)A taste of geometric quantization. Kahler manifolds.If you find errors in these notes, please email me! I thank Sheeyun Park and Curtis

    Vinson for catching lots of errors.

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    Contents

    1 From Newtons Laws to Langranges Equations 2

    1.1 Lagrangian and Newtonian Approaches . . . . . . . . . . . . . . . . . . . . 21.1.1 Lagrangian versus Hamiltonian Approaches . . . . . . . . . . . . . 61.2 Prehistory of the Lagrangian Approach . . . . . . . . . . . . . . . . . . . . 6

    1.2.1 The Principle of Minimum Energy . . . . . . . . . . . . . . . . . . 71.2.2 DAlemberts Principle and Lagranges Equations . . . . . . . . . . 81.2.3 The Principle of Least Time . . . . . . . . . . . . . . . . . . . . . . 101.2.4 How DAlembert and Others Got to the Truth . . . . . . . . . . . . 12

    2 Equations of Motion 14

    2.1 The Euler-Lagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

    2.1.2 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Interpretation of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    3 Lagrangians and Noethers Theorem 20

    3.1 Time Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.1 Canonical and Generalized Coordinates . . . . . . . . . . . . . . . . 21

    3.2 Symmetry and Noethers Theorem . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Noethers Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

    3.3 Conserved Quantities from Symmetries . . . . . . . . . . . . . . . . . . . . 243.3.1 Time Translation Symmetry . . . . . . . . . . . . . . . . . . . . . . 253.3.2 Space Translation Symmetry . . . . . . . . . . . . . . . . . . . . . . 253.3.3 Rotational Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 26

    3.4 Example Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 The Atwood Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.2 Disk Pulled by Falling Mass . . . . . . . . . . . . . . . . . . . . . . 293.4.3 Free Particle in Special Relativity . . . . . . . . . . . . . . . . . . . 31

    3.5 Electrodynamics and Relativistic Lagrangians . . . . . . . . . . . . . . . . 353.5.1 Gauge Symmetry and Relativistic Hamiltonian . . . . . . . . . . . . 353.5.2 Relativistic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 36

    v

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    CONTENTS 1

    3.6 Relativistic Particle in an Electromagnetic Field . . . . . . . . . . . . . . . 37

    3.7 Alternative Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.7.1 Lagrangian for a String . . . . . . . . . . . . . . . . . . . . . . . . . 393.7.2 Alternate Lagrangian for Relativistic Electrodynamics . . . . . . . . 41

    3.8 The General Relativistic Particle . . . . . . . . . . . . . . . . . . . . . . . 433.8.1 Free Particle Lagrangian in GR . . . . . . . . . . . . . . . . . . . . 443.8.2 Charged particle in EM Field in GR . . . . . . . . . . . . . . . . . 45

    3.9 The Principle of Least Action and Geodesics . . . . . . . . . . . . . . . . . 463.9.1 Jacobi and Least Time vs Least Action . . . . . . . . . . . . . . . . 463.9.2 The Ubiquity of Geodesic Motion . . . . . . . . . . . . . . . . . . . 48

    4 From Lagrangians to Hamiltonians 51

    4.1 The Hamiltonian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 514.2 Regular and Strongly Regular Lagrangians . . . . . . . . . . . . . . . . . . 54

    4.2.1 Example: A Particle in a Riemannian Manifold with Potential V(q) 544.2.2 Example: General Relativistic Particle in an E-M Potential . . . . . 554.2.3 Example: Free General Relativistic Particle with Reparameteriza-

    tion Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.4 Example: A Regular but not Strongly Regular Lagrangian . . . . . 55

    4.3 Hamiltons Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.1 Hamilton and Euler-Lagrange . . . . . . . . . . . . . . . . . . . . . 574.3.2 Hamiltons Equations from the Principle of Least Action . . . . . . 59

    4.4 Waves versus ParticlesThe Hamilton-Jacobi Equations . . . . . . . . . . 614.4.1 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4.2 The Hamilton-Jacobi Equations . . . . . . . . . . . . . . . . . . . . 63

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    Chapter 1

    From Newtons Laws to Langranges

    Equations

    (Week 1, March 28, 30, April 1.)