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Classical Dynamics
University of Cambridge Part II Mathematical TriposDr David Tong Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences,
Wilberforce Road,
Cambridge, CB3 OBA, UK
http://www.damtp.cam.ac.uk/user/tong/dynamics.htmld.tong@damtp.cam.ac.uk { 1 {
Recommended Books and Resources
L. Hand and J. Finch,Analytical Mechanics This very readable book covers everything in the course at the right level. It is similar to Goldstein's book in its approach but with clearer explanations, albeit at the expense of less content.
There are also three classic texts on the subject
H. Goldstein, C. Poole and J. Safko,Classical Mechanics In previous editions it was known simply as \Goldstein" and has been the canonical choice for generations of students. Although somewhat verbose, it is considered the standard reference on the subject. Goldstein died and the current, third, edition found two extra authors. L. Landau an E. Lifshitz,Mechanics This is a gorgeous, concise and elegant summary of the course in 150 content packed pages. Landau is one of the most important physicists of the 20th century and this is the rst volume in a series of ten, considered by him to be the \theoretical minimum" amount of knowledge required to embark on research in physics. In 30 years, only 43 people passed Landau's exam! A little known fact: Landau originally co-authored this book with one of his students, Leonid Pyatigorsky. They subsequently had a falling out and the authorship was changed. There are rumours that Pyatigorsky got his own back by denouncing Landau to the Soviet authorities, resulting in his arrest. V. I. Arnold,Mathematical Methods of Classical Mechanics Arnold presents a more modern mathematical approach to the topics of this course, making connections with the dierential geometry of manifolds and forms. It kicks o with \The Universe is an Ane Space" and proceeds from there...
Contents
1. Newton's Laws of Motion1
1.1Introduction1
1.2Newtonian Mechanics: A Single Particle2
1.2.1Angular Momentum3
1.2.2Conservation Laws4
1.2.3Energy4
1.2.4Examples5
1.3Newtonian Mechanics: Many Particles5
1.3.1Momentum Revisited6
1.3.2Energy Revisited8
1.3.3An Example9
2. The Lagrangian Formalism10
2.1The Principle of Least Action10
2.2Changing Coordinate Systems13
2.2.1Example: Rotating Coordinate Systems14
2.2.2Example: Hyperbolic Coordinates16
2.3Constraints and Generalised Coordinates17
2.3.1Holonomic Constraints18
2.3.2Non-Holonomic Constraints20
2.3.3Summary21
2.3.4Joseph-Louis Lagrange (1736-1813)22
2.4Noether's Theorem and Symmetries23
2.4.1Noether's Theorem24
2.5Applications26
2.5.1Bead on a Rotating Hoop26
2.5.2Double Pendulum28
2.5.3Spherical Pendulum29
2.5.4Two Body Problem31
2.5.5Restricted Three Body Problem33
2.5.6Purely Kinetic Lagrangians36
2.5.7Particles in Electromagnetic Fields36
2.6Small Oscillations and Stability38
2.6.1Example: The Double Pendulum41
{ 1 {
2.6.2Example: The Linear Triatomic Molecule42
3. The Motion of Rigid Bodies45
3.1Kinematics46
3.1.1Angular Velocity47
3.1.2Path Ordered Exponentials49
3.2The Inertia Tensor50
3.2.1Parallel Axis Theorem52
3.2.2Angular Momentum53
3.3Euler's Equations53
3.3.1Euler's Equations54
3.4Free Tops55
3.4.1The Symmetric Top55
3.4.2Example: The Earth's Wobble57
3.4.3The Asymmetric Top: Stability57
3.4.4The Asymmetric Top: Poinsot Construction58
3.5Euler's Angles62
3.5.1Leonhard Euler (1707-1783)64
3.5.2Angular Velocity65
3.5.3The Free Symmetric Top Revisited65
3.6The Heavy Symmetric Top67
3.6.1Letting the Top Go70
3.6.2Uniform Precession71
3.6.3The Sleeping Top72
3.6.4The Precession of the Equinox72
3.7The Motion of Deformable Bodies74
3.7.1Kinematics74
3.7.2Dynamics77
4. The Hamiltonian Formalism80
4.1Hamilton's Equations80
4.1.1The Legendre Transform82
4.1.2Hamilton's Equations83
4.1.3Examples84
4.1.4Some Conservation Laws86
4.1.5The Principle of Least Action87
4.1.6What's Your Name, Man? William Rowan Hamilton (1805-1865)88
4.2Liouville's Theorem88
{ 2 {
4.2.1Liouville's Equation90
4.2.2Time Independent Distributions91
4.2.3Poincare Recurrence Theorem92
4.3Poisson Brackets94
4.3.1An Example: Angular Momentum and Runge-Lenz95
4.3.2An Example: Magnetic Monopoles96
4.3.3An Example: The Motion of Vortices98
4.4Canonical Transformations100
4.4.1Innitesimal Canonical Transformations102
4.4.2Noether's Theorem Revisited104
4.4.3Generating Functions104
4.5Action-Angle Variables105
4.5.1The Simple Harmonic Oscillator105
4.5.2Integrable Systems107
4.5.3Action-Angle Variables for 1d Systems108
4.5.4Action-Angle Variables for the Kepler Problem111
4.6Adiabatic Invariants113
4.6.1Adiabatic Invariants and Liouville's Theorem116
4.6.2An Application: A Particle in a Magnetic Field116
4.6.3Hannay's Angle118
4.7The Hamilton-Jacobi Equation121
4.7.1Action and Angles from Hamilton-Jacobi124
4.8Quantum Mechanics126
4.8.1Hamilton, Jacobi, Schrodinger and Feynman128
4.8.2Nambu Brackets131
{ 3 {
Acknowledgements
These notes rely heavily on the textbooks listed at the beginning and on notes from past courses given by others, in particular Anne Davis, Gary Gibbons, Robin Hud- son, Michael Peskin and Neil Turok. My thanks also to Michael Efroimsky and Matt Headrick for useful comments. I am supported by the Royal Society. { 4 {
1. Newton's Laws of Motion
\So few went to hear him, and fewer understood him, that oftimes he did, for want of hearers, read to the walls. He usually stayed about half an hour; when he had no auditors he commonly returned in a quarter of that time." Appraisal of a Cambridge lecturer in classical mechanics, circa 1690
1.1 Introduction
The fundamental principles of classical mechanics were laid down by Galileo and New- ton in the 16 thand 17thcenturies. In 1686, Newton wrote thePrincipiawhere he gave us three laws of motion, one law of gravity and pretended he didn't know cal- culus. Probably the single greatest scientic achievement in history, you might think this pretty much wraps it up for classical mechanics. And, in a sense, it does. Given a collection of particles, acted upon by a collection of forces, you have to draw a nice diagram, with the particles as points and the forces as arrows. The forces are then added up and Newton's famous \F=ma" is employed to gure out where the par- ticle's velocities are heading next. All you need is enough patience and a big enough computer and you're done. From a modern perspective this is a little unsatisfactory on several levels: it's messy and inelegant; it's hard to deal with problems that involve extended objects rather than point particles; it obscures certain features of dynamics so that concepts such as chaos theory took over 200 years to discover; and it's not at all clear what the relationship is between Newton's classical laws and quantum physics. The purpose of this course is to resolve these issues by presenting new perspectives on Newton's ideas. We shall describe the advances that took place during the 150 years after Newton when the laws of motion were reformulated using more powerful techniques and ideas developed by some of the giants of mathematical physics: people such as Euler, Lagrange, Hamilton and Jacobi. This will give us an immediate practical advantage, allowing us to solve certain complicated problems with relative ease (the strange motion of spinning tops is a good example). But, perhaps more importantly, it will provide an elegant viewpoint from which we'll see the profound basic principles which underlie Newton's familiar laws of motion. We shall prise open \F=ma" to reveal the structures and symmetries that lie beneath. { 1 { Moreover, the formalisms that we'll develop here are the basis forallof fundamental modern physics. Every theory of Nature, from electromagnetism and general relativity, to the standard model of particle physics and more speculative pursuits such as string theory, is best described in the language we shall develop in this course. The new formalisms that we'll see here also provide the bridge between the classical world and the quantum world. There are phenomena in Nature for which these formalisms are not particularly useful. Systems which are dissipative, for example, are not so well suited to these new techniques. But if you want to understand the dynamics of planets and stars and galaxies as they orbit and spin, or you want to understand what's happening at the LHC where protons are collided at unprecedented energies, or you want to know how electrons meld together in solids to form new states of matter, then the foundations that we'll lay in in this course are a must.
1.2 Newtonian Mechanics: A Single Particle
In the rest of this section, we'll take a
ying tour through the basic ideas of classical mechanics handed down to us by Newton. More details can be found in thelectures on Dynamics and Relativity. We'll start with a single particle.. Aparticleis dened to be an object of insignicant size. e.g. an electron, a tennis ball or a planet. Obviously the validity of this statement depends on the context: to rst approximation, the earth can be treated as a particle when computing its orbit around the sun. But if you want to understand its spin, it must be treated as an extended object. The motion of a particle of massmat the positionris governed byNewton's Second
LawF=maor, more precisely,
F(r;_r) =_p(1.1)
whereFis the force which, in general, can depend on both the positionras well as the velocity _r(for example, friction forces depend on_r) andp=m_ris the momentum. BothFandpare 3-vectors which we denote by the bold font. Equation (1.1) reduces toF=maif _m= 0. But ifm=m(t) (e.g. in rocket science) then the form with_pis correct. General theorems governing dierential equations guarantee that if we are givenr and _rat an initial timet=t0, we can integrate equation (1.1) to determiner(t) for all t(as long asFremains nite). This is the goal of classical dynamics. { 2 { Equation (1.1) is not quite correct as stated: we must add the caveat that it holds only in aninertial frame. This is dened to be a frame in which a free particle with _m= 0 travels in a straight line, r=r0+vt(1.2) Newtons's rst lawis the statement that such frames exist. An inertial frame is not unique. In fact, there are an innite number of inertial frames. LetSbe an inertial frame. Then there are 10 linearly independent transformations S!S0such thatS0is also an inertial frame (i.e. if (1.2) holds inS, then it also holds inS0). These are 3 Rotations:r0=OrwhereOis a 33 orthogonal matrix. 3 Translations:r0=r+cfor a constant vectorc. 3 Boosts:r0=r+utfor a constant velocityu. 1 Time Translation:t0=t+cfor a constant real numberc If motion is uniform inS, it will also be uniform inS0. These transformations make up theGalilean Groupunder which Newton's laws are invariant. They will be impor- tant in section 2.4 where we will see that these symmetries of space and time are the underlying reason for conservation laws. As a parenthetical remark, recall fromspecial relativitythat Einstein's laws of motion are invariant under Lorentz transformations which, together with translations, make up the Poincare group. We can recover the Galilean group from the Poincare group by taking the speed of light to innity.
1.2.1 Angular Momentum
We dene theangular momentumLof a particle and thetorqueacting upon it as
L=rp;=rF(1.3)
Note that, unlike linear momentump, bothLanddepend on where we take the origin: we measure angular momentum with respect to a particular point. Let us cross both sides of equation (1.1) withr. Using the fact that_ris parallel top, we can write ddt (rp) =r_p. Then we get a version of Newton's second law that holds for angular momentum: =_L(1.4) { 3 {
1.2.2 Conservation Laws
From (1.1) and (1.4), two important conservation laws follow immediately. IfF= 0 thenpis constant throughout the motion If= 0 thenLis constant throughout the motion Notice that= 0 does not requireF= 0, but onlyrF= 0. This means thatF must be parallel tor. This is the denition of acentral force. An example is given by the gravitational force between the earth and the sun: the earth's angular momentum about the sun is constant. As written above in terms of forces and torques, these conservation laws appear trivial. In section 2.4, we'll see how they arise as a property of the symmetry of space as encoded in the Galilean group.
1.2.3 Energy
Let's now recall the denitions of energy. We rstly dene thekinetic energyTas T=12 m_r_r(1.5) Suppose from now on that the mass is constant. We can compute the change of kinetic energy with time: dTdt =_p_r=F_r. If the particle travels from positionr1at timet1 to positionr2at timet2then this change in kinetic energy is given by
T(t2) T(t1) =Z
t2 t 1dTdt dt=Z t2 t
1F_rdt=Z
r2 r
1Fdr(1.6)
where the nal expression involving the integral of the force over the path is called the work doneby the force. So we see that the work done is equal to the change in kinetic energy. From now on we will mostly focus on a very special type of force known as a conservativeforce. Such a force depends only on positionrrather than velocity_rand is such that the work done is independent of the path taken. In particular, for a closed path, the work done vanishes. I
Fdr= 0, r F= 0 (1.7)
It is a deep property of
at spaceR3that this property implies we may write the force as
F= rV(r) (1.8)
for somepotentialV(r). Systems which admit a potential of this form include gravi- tational, electrostatic and interatomic forces. When we have a conservative force, we { 4 { necessarily have a conservation law for energy. To see this, return to equation (1.6) which now reads
T(t2) T(t1) = Z
r2 r
1rVdr= V(t2) +V(t1) (1.9)
or, rearranging things,
T(t1) +V(t1) =T(t2) +V(t2)E(1.10)
SoE=T+Vis also a constant of motion. It is the energy. When the energy is considered to be a function of positionrand momentumpit is referred to as the HamiltonianH. In section 4 we will be seeing much more of the Hamiltonian.
1.2.4 Examples
Example 1: The Simple Harmonic Oscillator This is a one-dimensional system with a force proportional to the distancexto the origin:F(x) = kx. This force arises from a potentialV=12 kx2. SinceF6= 0, momentum is not conserved (the object oscillates backwards and forwards) and, since the system lives in only one dimension, angular momentum is not dened. But energy E=12 m_x2+12 kx2is conserved. Example 2: The Damped Simple Harmonic Oscillator We now include a friction term so thatF(x;_x) = kx _x. SinceFis not conservative, energy is not conserved. This system loses energy until it comes to rest. Example 3: Particle Moving Under Gravity Consider a particle of massmmoving in 3 dimensions under the gravitational pull of a much larger particle of massM. The force isF= (GMm=r2)^rwhich arises from the potentialV= GMm=r. Again, the linear momentumpof the smaller particle is not conserved, but the force is both central and conservative, ensuring the particle's total energyEand the angular momentumLare conserved.
1.3 Newtonian Mechanics: Many Particles
It's easy to generalise the above discussion to many particles: we simply add an index to everything in sight! Let particleihave massmiand positionriwherei= 1;:::;N is the number of particles. Newton's law now reads F i=_pi(1.11) { 5 { whereFiis the force on theithparticle. The subtlety is that forces can now be working between particles. In general, we can decompose the force in the following way: F i=X j6=iF ij+Fexti(1.12) whereFijis the force acting on theithparticle due to thejthparticle, whileFextiis the external force on theithparticle. We now sum over allNparticles X iF i=X i;jwithj6=iF ij+X iF exti = X i