Preprint typeset in JHEP style - PAPER VERSION 402-2203-01L, PHY311
MechanicsETH & University of Zurich, Wintersemester 2009
Prof. Dr. Ben Moore
Institute for Theoretical PhysicsUniversity of ZurichWinterthurerstrasse 190Irchel, Zurich
Office 36K72Telephone 044 635 5815www.itp.uzh.chwww.itp.uzh.ch/∼moore/[email protected]
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Abstract: This course on ”Analytical Mechanics” will cover the following topics
with various diversions along the way:
Introduction, history, context
Review of Newton’s Laws of Motion
N-particle systems
D’Alemberts principle
Hamilton’s principle of least action
Lagrangian Mechanics
Orbits in two and three body systems
Noether’s Theorem and Symmetries
Motion of Rigid Bodies
Hamiltonian Mechanics
Configuration Space and Phase Space
Canonical Transformations
Action-Angle Variables
Hamilton-Jacobi formalism
Liouville’s Theorem
Chaotic dynamics, integrable and non-integrable systems
Who is your lecturer? I am director of the Institute for Theoretical Physics (ITP) at
the University of Zurich. I started the theoretical and computational astrophysics and
cosmology group and we carry out research on a range of topics from the formation
of planetary systems, galaxy formation, dark matter and dark energy, to the origin
and future of the universe. To simulate the universe we use the worlds largest super-
computers as well as the zbox supercomputer on Irchel campus which was designed
and constructed by our group.
1. Text Books
There are many classic text books on mechanics, some are listed below. You don’t
need to buy a book since we will cover all the material you need to know in class.
However it would certainly help if you do have access to one or more textbooks. If
you want to buy one book, I recommend either Goldstein or Fasano. If you want to
supplement the lecture notes with more advanced material I recommend the graduate
level texts by Arnold or McCauley.
Course notes, homework problems and solutions will be posted on the course
website: www.itp.uzh.ch/∼moore/mechanics2009
I will try to post the notes for each lecture the day before each lecture takes
place.
Some material (about one third) including example problems and computer
demonstrations of chaos or orbits etc will be covered only on the blackboard or
using the beamer.
• L. Hand and J. Finch, Analytical Mechanics
This is quite a good readable book and 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. On the downside there are far too many ”reader
should solve” questions in the text.
• C. Lanczos, The Variational Principles of Mechanics
Lanczos was famous for his work in GR and Mathematics. This book was first
published in 1949 and updated in 1970 before he died. It is a very readable account
of analytical mechanics at an intermediate level. This text contains very interesting
historical and philosophical discussions.
• S. Thornton and J. Marion, Classical dynamics of particles and systems
Too low level for this course, but good if you have to catch up. This book covers the
basics of mechanics very well, but does not go to a high enough level for this course.
• 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. The text is too long (over 600 pages), however
it is considered the standard reference on the subject. Since Goldstein died newer
editions found two extra authors.
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• J. McCauley, Classical Mechanics
Excellent textbook, but very advanced. This is mechanics at the highest level. The
book starts with a nice historical overview and then the focus is on integrability
and chaotic dynamics. If you enjoy mechanics and chaos theory or perhaps wish to
continue in research then this is a good book to buy.
• L. Landau and E. Lifshitz, Mechanics
This is an amazingly concise and elegant summary of mechanics in 150 content
packed pages. However the text is very difficult to work through with many steps
of the derivations missing. Landau is one of the most important physicists of the
20th century and this is the first 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!
• V. I. Arnold, Mathematical Methods of Classical Mechanics
Advanced mathematical and geometrical treatment of mechanics. Arnold presents a
more modern mathematical approach to the topics of this course, making connections
with the differential geometry of manifolds and forms. It kicks off with “The Uni-
verse is an Affine Space” and proceeds from there... A wonderful book for advanced
readers.
• A. Fasano and S. Marmi, Analytical Mechanics
The most recent textbook in this list, published in 2002. An excellent 750 page book
that explains mechanics starting with a geometrical view of the world. A tough
choice between this and Goldstein as the one book you might own. Contains many
modern problems, applications and examples.
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2. First things first
2.1 How to learn
It took over 150 years to develop Hamiltonian dynamics, you are not going to learn
it by studying the night before solutions are due, or the few days before the exams.
If you like, and this often works well, work in small groups of 2-6 people. When you
are solving problems, make sure you understand what you are doing. You should be
able to explain the solutions to the problems to your colleagues.
There is no point in simply copying solutions to problems from textbooks or from
other students - they do not count toward the final grade. Attempting the problems
is the only way you can learn mechanics. Reading the notes will not suffice since it
does not give you a deep enough understanding of the subject. I cannot emphasise
strongly enough, that in addition to reading and understanding the lecture notes, you
should attempt the problems and understanding the solutions. Ideally, you would
supplement these notes with a textbook at the level of Goldstein.
2.2 Course requirements
• Attempt and hand in the problems. Each problem will be graded 1 (mostly
incorrect), 2 (good attempt) or 3 (mostly correct). You should achieve over
65% in order to qualify to sit the final exam.
• Take the multiple choice mid-term exam during the course. A poor mid-term
score will not move your final grade downwards.
• The final grade will be based on a 3 hour written exam in 2010 with about
seven problems, some with multiple parts.
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2.3 Problems classes
At the end of the lecture today, the ETH students should come to the front and sign
up for a given problems class. There are four of these at 10:15-12:00 on Wednesdays
and one at 8:15-10:00 on Tuesday (primarily for the Math students who can’t attend
the Wednesday class). Homework problems will be handed out during the Monday
lectures.
• The ETH students will be given hints to the problems during the next problem
class, followed by solutions to the previous weeks problems. They should hand
in solutions in the appropriate boxes in building HIT by 4pm the following
Monday.
• The University students will be given hints to the problems during their prob-
lem class on Fridays (10:15 at the ITP) followed by solutions to the previous
weeks problems. They should hand in solutions in the appropriate boxes at the
Theoretical Physics Institute, building 36 Irchel by 4pm the following Wednes-
day.
Any questions?
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3. History & Context
”Wise men speak because they have something to say; Fools because they
have to say something.” (Plato)
”I have never met a man so ignorant that I couldn’t learn something from
him.” (Galileo)
3.1 Euclid to Galileo
Euclid (ca. 300 BC) to Archimedes (287 BC) marked the beginnings of geometry,
algebra and statics. The subsequent era in Western Europe, from later Greek and
Roman times through until the 13th century produced little of scientific or mathe-
matical value. The reason for this was the adoption by Western Christianity of the
ideas of Plato and his successor Aristotle (ca. 400BC). Plato developed an abstract
model cosmology of the heavens that was fundamentally anti-empirical since it was
based on ideas of motion that grouped together unrelated phenomenon such as a
rolling ball, the growth of an acorn etc.
One reason for the stagnation of European science was due to the Roman edu-
cation system which emphasised law, obedience etc, and the mathematical writings
from the libraries of Alexandria were not passed on or made available to Western
medieval Europe. In the first few centuries Islamic conquests left ”academic centres”
such as Alexandria and Syracuse cut off from the West. One of the first libraries
north of the Alps was due to the Irish established in St. Gallen.
Activity increased around the 12th century - the recursive reasoning of Euclid
(used to perform the division of two numbers), was re-introduced by Fibonacci. Such
work was near impossible under the Roman number system: MDCLXVI - XXIX =
MDCXXXVII !!! Euclids work was finally translated into Latin around 500AD,
roughly the time of the unification of the Frankish tribes after they conquered the
Roman Celtic province of Gaul. (n.b. it was the Frankish domination of Europe
and the Northern Alps that led to the Lansgemeinde in the three Swiss cantons -
common grazing rights and local justice, as opposed to private property and abstract
justice of the Romans.)
The mathematical and scientific revolution began in earnest by the 17th century
in Northern Europe at a time when conflict with the church was common place.
Galileo (1554-1642) was placed under house arrest in Northern Italy. Descartes (1596-
1650) and Newton (1642-1727) were Deists (”One who believes in the existence of a
God or supreme being but denies revealed religion, basing his belief on the light of
nature and reason”). At this time, the Anglican church in Virginia, USA still burned
heretics at the stake!
Universal mechanistic laws of nature were established in the age of Kepler,
Galileo and Descartes. Finally Plato’s ideas were overturned. Plato was obsessed
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with symmetry and perfection. He taught that circular motion at constant speed was
a natural consequence of nature. If empirical observations disagreed with this then
he argued that the observations must be wrong (although some modern scientists
still follow this approach!). His Earth-centric model of the solar system, in which
the sun and planets move on circular orbits around the Earth, was in conflict with
the apparent retrograde motion of the planets at certain times of the year. Ptolemy
(100-170AD) added epicycles (circular corrections) to the uniform motion to try to
save the ideas of Plato. Aristarchus (310BC) had believed that the sun was at the
centre of the solar system, yet this idea lay dormant until the Polish monk Coperni-
cus advanced it strongly enough in the 16th century such that it made the Vaticans
best seller list (”the index”).
In neo-Platonic christianity, the planets moved on perfect solid spheres advanced
by angels and spirits, based on the ideas of Aristotle that whatever moves must be
moved, such that constant motion required a constant force. Angels and spirits
must be invoked for the simplest phenomenon such as a thrown stone which moves
until it hits the ground. In Aristolian physics, weight was an inherent property of
an object and the natural state of objects was to be at rest. This was based on
naive observation and lack of rigorous experimental checks. The law of inertia and
Newton’s second law of motion are neither intuitive or obvious, if they were then
persistent thinkers such as Plato and Aristotle might have discovered the correct
mathematical laws of nature a long time ago.
Not long before Newton, algebra was stated in words. The ideas of writing
symbols to represent unknown quantities and irrational numbers were not developed
until Descartes and Fermat. Fifty years before Newton sufficient mathematics were
developed such that he could invent differential calculus and dynamics. Newton’s
unified theory of mechanics was based on the building blocks of the two local laws
of theoretical dynamics discovered empirically by Galileo and the global laws of two
body motion by Kepler. The fundamental idea of universal mechanistic laws of
nature was advocated by Descartes and realised by Newton.
Tycho Brahe spent most of his life observing and making notes of the motions
of the planets. (Although he also observed the appearance of a new star, as well as a
comet, which indicated that the heavens were not invariant, an observation ignored
by the church at that time). Kepler in 1577 started to work on Brahe’s data for
Mars, which showed the largest deviations from a non-circular orbit. After much
effort he showed that Mars lies on a closed (precisely repeating), but non-circular
orbit, an ellipse with one focus at the sun. In spite of the lack of motion about the
sun at a constant speed, he showed that Mars traced out equal areas in equal times.
Thought problem: If the motion of a planet in a central potential (provided by the
sun) did not give closed regular orbits, what would be the implications for science?
For life on Earth?
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3.2 On the shoulders of giants
Around this time, Galileo was performing controlled reproducible experiments on the
motion of bodies to infer that the vertical acceleration
ay = −g (3.1)
where g ≈ 9.8m/s2. This new ”universal law of nature” violated Aristotle’s notions
of weight. Galileo also discovered the fundamental concept of inertia such that for
horizontal motion the velocity
vx = constant. (3.2)
Descartes generalised this to assert that the natural state of motion of an undisturbed
body in the universe is not rest, or circular motion, but rectilinear motion at constant
speed. This was the foundation of Physics! Without this law, Newton’s 2nd law
would not follow. Even though Kepler sent his results to Galileo, for some reason
these were ignored by Galileo and there was no unification of their separate results
and ideas... Until Newton.
Following Newton, the law of inertia can be written dv/dt = 0 where v is the
body’s unperturbed velocity. If such force free motion occurs at a constant velocity
then how does one describe motion that differs from this natural state? The simplest
generalisation of Galileo’s local laws (3.1 and 3.2) is
mdv
dt=−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→cause of deviation from uniform rectilinear motion (3.3)
where m is a coefficient of resistance to and change in velocity. (Newton could have
introduced a tensor coefficient mij but he implicitly assumed that Euclidean space
is isotropic.) Thus Newton finally understood that it is the acceleration, not the
velocity, that is directly connected to the idea of force. Thus Newton could write his
2nd law
dp/dt = F (3.4)
and if the coefficient of inertia m = const then F = ma.
If we follow Newton, then the net force that alters the motion of a body can
only be due to the presence of other bodies whose motions are temporarily ignored
in equation (3.4), bodies whose own accelerations must also obey (3.4). This idea
is represented by the action-reaction principle Fij = −Fji, in other words, forces are
not caused by properties of empty space.
So begins the story of Mechanics.
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Figure 1: Some famous dynamicists, many with beards...
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4. Newton’s Laws of Motion
Mechanics began...
“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
”If I have seen farther than (others) Descartes, it is because I was standing
on the shoulders of giants.”
4.1 Introduction
The fundamental principles of classical mechanics were laid down by Galileo and
Newton in the 16th and 17th centuries. In 1686, Newton published the Principia
in which he wrote down three laws of motion, one law of gravity and pretended he
didn’t know calculus. Probably the single greatest scientific 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 figure out
where the particle’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.
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Moreover, the formalisms that we’ll develop here are the basis for all of funda-
mental 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 know what will happen when
we eventually turn on the next particle collider and smash protons together at un-
precedented energies and watch new particles come tumbling out; 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.
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Figure 2: Complexity versus number of variables
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4.2 Newtonian Mechanics: A Single Particle
In the rest of this section, we’ll take a quick tour through the basic ideas of classical
mechanics handed down to us by Newton. We’ll start with a single particle.
A particle is defined to be an object of insignificant size. e.g. an electron, a tennis
ball or a planet. Obviously the validity of this statement depends on the context: to
first approximation, the Earth can be treated as a particle when computing its orbit
around the sun. But if you want to understand why it wobbles as its spins, it must
be treated as an extended object.
The motion of a particle of mass m at the position r is governed by Newton’s
Second Law F = ma or, more precisely,
F(r, r) = p (4.1)
where F is the force which, in general, can depend on both the position r as well as the
velocity r (for example, friction forces depend on r) and p = mr is the momentum.
Both F and p are 3-vectors which we denote by the bold font. Equation (4.1) reduces
to F = ma if m = 0. But if m = m(t) (e.g. in rocket science) then the form with p
is correct.
General theorems governing differential equations guarantee that if we are given r
and r at an initial time t = t0, we can integrate equation (4.1) to determine r(t) for
all t (as long as F remains finite). This is the goal of classical dynamics.
Equation (4.1) is not quite correct as stated: we must add the caveat that it holds
only in an inertial frame. This is defined to be a frame in which a free particle with
m = 0 travels in a straight line,
r = r0 + vt (4.2)
Newtons’s first law is the statement that such frames exist.
An inertial frame is not unique. In fact, there are an infinite number of inertial
frames. Let S be an inertial frame. Then there are 10 linearly independent transfor-
mations S → S ′ such that S ′ is also an inertial frame (i.e. if (4.2) holds in S, then
it also holds in S ′). These are
• 3 Rotations: r′ = Or where O is a 3× 3 orthogonal matrix.
• 3 Translations: r′ = r + c for a constant vector c.
• 3 Boosts: r′ = r + ut for a constant velocity u.
• 1 Time Translation: t′ = t+ c for a constant real number c
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If motion is uniform in S, it will also be uniform in S ′. These transformations make up
the Galilean Group under which Newton’s laws are invariant. They will be important
later when we will see that these symmetries of space and time are the underlying
reason for conservation laws. As a parenthetical remark, recall from special relativity
that 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 infinity.
4.2.1 Angular Momentum
We define the angular momentum L of a particle and the torque τ acting upon it as
L = r× p , τ = r× F (4.3)
Note that, unlike linear momentum p, both L and τ depend on where we take the
origin: we measure angular momentum with respect to a particular point. Let us
cross both sides of equation (4.1) with r. Using the fact that r is parallel to p, we
can write ddt
(r × p) = r × p. Then we get a version of Newton’s second law that
holds for angular momentum:
τ = L (4.4)
4.2.2 Conservation Laws
From (4.1) and (4.4), two important conservation laws follow immediately.
• If F = 0 then p is constant throughout the motion
• If τ = 0 then L is constant throughout the motion
Notice that τ = 0 does not require F = 0, but only r × F = 0. This means that
F must be parallel to r. This is the definition of a central 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. Later, we’ll see how they arise as a
property of the symmetry of space as encoded in the Galilean group.
4.2.3 Energy
Let’s now recall the definitions of energy. We firstly define the kinetic energy T as
T = 12m r · r (4.5)
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 position r1
at time t1 to position r2 at time t2 then this change in kinetic energy is given by
T (t2)− T (t1) =
∫ t2
t1
dT
dt=
∫ t2
t1
F · r dt =
∫ r2
r1
F · dr (4.6)
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where the final expression involving the integral of the force over the path is called
the work done by 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 conservative force. Such a force depends only on position r rather than
velocity r and is such that the work done is independent of the path taken. In
particular, for a closed path, the work done vanishes.∮F · dr = 0 ⇔ ∇× F = 0 (4.7)
It is a deep property of flat space R3 that this property implies we may write the
force as
F = −∇V (r) (4.8)
for some potential V (r). Systems which admit a potential of this form include gravi-
tational, electrostatic and interatomic forces. When we have a conservative force, we
necessarily have a conservation law for energy. To see this, return to equation (4.6)
which now reads
T (t2)− T (t1) = −∫ r2
r1
∇V · dr = −V (t2) + V (t1) (4.9)
or, rearranging things,
T (t1) + V (t1) = T (t2) + V (t2) ≡ E (4.10)
So E = T + V is also a constant of motion. It is the energy. When the energy is
considered to be a function of position r and momentum p it is referred to as the
Hamiltonian H. Later we will be seeing much more of the Hamiltonian.
4.2.4 Examples
• Example 1: The Simple Harmonic Oscillator
This is a one-dimensional system with a force proportional to the distance x to the
origin: F (x) = −kx. This force arises from a potential V = 12kx2. Since F 6= 0,
momentum is not conserved (the object oscillates backwards and forwards) and,
since the system lives in only one dimension, angular momentum is not defined. But
energy E = 12mx2 + 1
2kx2 is conserved.
• Example 2: The Damped Simple Harmonic Oscillator
We now include a friction term so that F (x, x) = −kx− γx. Since F is not conser-
vative, energy is not conserved. This system loses energy until it comes to rest.
• Example 3: Particle Moving Under Gravity
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Consider a particle of mass m moving in 3 dimensions under the gravitational pull of
a much larger particle of mass M . The force is F = GMm/r2r which arises from the
potential V = −GMm/r. Again, the linear momentum p of the smaller particle is
not conserved, but the force is both central and conservative, ensuring the particle’s
total energy E and the angular momentum L are conserved.
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