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.)

    Classical mechanics is a very peculiar branch of physics. It used to be considered thesum total of our theoretical knowledge of the physical universe (Laplaces daemon, theNewtonian clockwork), but now it is known as an idealization, a toy model if you will.The astounding thing is that probably all professional applied physicists still use classicalmechanics. So it is still an indispensable part of any physicists or engineers education.

    It is so useful because the more accurate theories that we know of (general relativity

    and quantum mechanics) make corrections to classical mechanics generally only in extremesituations (black holes, neutron stars, atomic structure, superconductivity, and so forth).Given that GR and QM are much harder theories to use and apply it is no wonder thatscientists will revert to classical mechanics whenever possible.

    So, what is classical mechanics?

    1.1 Lagrangian and Newtonian Approaches

    We begin by comparing the Newtonian approach to mechanics to the subtler approach ofLagrangian mechanics. Recall Newtons law:

    F = ma (1.1)

    wherein we consider a particle moving in n. Its position, say q, depends on time t

    ,so it defines a function,

    q :

    n.

    From this function we can define velocity,

    v = q :

    n

    2

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    1.1 Lagrangian and Newtonian Approaches 3

    where q = dqdt

    , and also acceleration,

    a = q :

    n.

    Now let m > 0 be the mass of the particle, and let F be a vector field on n called the

    force. Newton claimed that the particle satisfies F = ma. That is:

    m a(t) = F(q(t)) . (1.2)

    This is a 2nd-order differential equation for q :

    n which will have a unique solution

    given some q(t0) and q(t0), provided the vector field F is nice by which we technicallymean smooth and bounded (i.e., |F(x)| < B for some B > 0, for all x

    n).We can then define a quantity called kinetic energy:

    K(t) :=1

    2m v(t) v(t) (1.3)

    This quantity is interesting because

    d

    dtK(t) = m v(t) a(t)

    = F(q(t)) v(t)

    So, kinetic energy goes up when you push an object in the direction of its velocity, andgoes down when you push it in the opposite direction. Moreover,

    K(t1) K(t0) =

    t1t0

    F(q(t)) v(t) dt

    =

    t1t0

    F(q(t)) q(t) dt

    So, the change of kinetic energy is equal to the work done by the force, that is, the integralof F along the curve q : [t0, t1]

    n. This implies (by Stokess theorem relating lineintegrals to surface integrals of the curl) that the change in kinetic energy K(t1) K(t0)is independent of the curve going from q(t0) = a to q(t1) = b iff

    F = 0.

    This in turn is true iffF = V (1.4)

    for some function V : n

    . This function is then unique up to an additive constant; wecall it the potential. A force with this property is called conservative. Why? Becausein this case we can define the total energy of the particle by

    E(t) := K(t) + V(q(t)) (1.5)

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    4 From Newtons Laws to Langranges Equations

    where V(t) := V(q(t)) is called the potential energy of the particle, and then we can

    show that E is conserved: that is, constant as a function of time. To see this, note thatF = ma implies

    d

    dt[K(t) + V(q(t))] = F(q(t)) v(t) + V(q(t)) v(t)

    = 0, (because F = V).

    Conservative forces let us apply a whole bunch of cool techniques. In the Lagrangianapproach we define a quantity

    L := K(t) V(q(t)) (1.6)

    called the Lagrangian, and for any curve q : [t0, t1]

    n with q(t0) = a, q(t1) = b, wedefine the action to be

    S(q) :=

    t1t0

    L(t) dt (1.7)

    From here one can go in two directions. One is to claim that nature causes particlesto follow paths of least action, and derive Newtons equations from that principle. Theother is to start with Newtons principles and find out what conditions, if any, on S(q)follow from this. We will use the shortcut of hindsight, bypass the philosophy, and simplyuse the mathematics of variational calculus to show that particles follow paths that arecritical points of the action S(q) if and only if Newtons law F = ma holds. To do this,

    n

    q

    qs + sq

    t0 t1

    Figure 1.1: A particle can sniff out the path of least action.

    let us look for curves (like the solid line in Fig. 1.1) that are critical points ofS, namely:

    d

    dsS(qs)|s=0 = 0 (1.8)

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    1.1 Lagrangian and Newtonian Approaches 5

    where

    qs = q + sqfor all q : [t0, t1]

    n with,q(t0) = q(t1) = 0.

    To show that

    F = ma d

    dsS(qs)|t=0 = 0 for all q : [t0, t1]

    n with q(t0) = q(t1) = 0 (1.9)

    we start by using integration by parts on the definition of the action, and first noting thatdqs/ds = q(t) is the variation in the path:

    d

    dsS(qs)

    s=0

    =d

    ds

    t1t0

    1

    2mqs(t) qs(t) V(qs(t)) dt

    s=0

    =

    t1t0

    d

    ds

    1

    2mqs(t) qs(t) V(qs(t))

    dt

    s=0

    =

    t1t0

    mqs

    d

    dsqs(t) V(qs(t))

    d

    dsqs(t)

    dt

    s=0

    Note thatd

    dsqs(t) =

    d

    ds

    d

    dtqs(t) =

    d

    dt

    d

    dsqs(t)

    and when we set s = 0 this quantity becomes just:

    d

    dtq(t)

    So,

    d

    dsS(qs)

    s=0

    =

    t1t0

    mq

    d

    dtq(t) V(q(t)) q(t)

    dt

    Then we can integrate by parts, noting the boundary terms vanish because q = 0 at t1and t0:

    d

    dsS(qs)|s=0 =

    t1t0

    [mq(t) V(q(t))] q(t)dt

    It follows that variation in the action is zero for all variations q iff the term in brackets

    is identically zero, that is, mq(t) V(q(t)) = 0

    So, the path q is a critical point of the action S iff

    F = ma.

    The above result applies only for conservative forces, i.e., forces that can be writtenas the minus the gradient of some potential. However, this seems to be true of the mostfundamental forces that we know of in our universe. It is a simplifying assumption thathas withstood the test of time and experiment.

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