Part IB Advanced Physics Course


1B Dynamics: 16 Lectures by David J.C. MacKay
From 1999 to 2001 inclusive I taught the 2ndyear Dynamics course.
In 2002 I will be supported by a Research Fellowship and will stop teaching
this course.
Questions about the course are answered here.

Any other questions?

I will hold a clinic on Tuesdays and Saturdays after lectures
in the Old Bursary, Darwin College.
Anyone is welcome to come along and give feedback or ask questions.
Physics teaching by the Inference group
is supported by the Gatsby charitable foundation.

Here is the official course synopsis, from the Physics Course Handbook.
Accompanying pages give more details on the lecture sequence and emphases of the course.
The energy method:
Equations of motion from energy functions.
Central force problems reexpressed as onedimensional problems.
Perturbations of
circular orbits.
Lagrangian and Hamiltonian dynamics.
Statespace diagrams.
Conservation of angular momentum, energy, phase space volume.
Perturbation methods and the simple pendulum. Dimensional analysis.
Normal modes: Counting degrees of freedom. Square matrices as linear operators
and in quadratic forms. Eigenvectors. Stability analysis using normal modes.
Use of symmetries to find normal modes. Beats. Modes of molecules.
Elasticity: Definitions of strain and stress as tensors. Young's
modulus, Poisson ratio, Shear and bulk modulus. Relationship
between shear, compression, and extension.
Central forces: Kepler's laws. Planetary orbits. Scattering.
Predicting cross sections using dimensional analysis. Orbital transfers.
Gravitational slingshot.
Rotating frames and fictitious forces: Centrifugal and Coriolis
forces.
Rigid bodies: Relationship between angular velocity vector
and angular momentum vector. Collisions of rigid bodies.
Precession of the earth.

The course text is:
Analytical Mechanics, Hand, L.N. and Finch, J.D. (Cambridge
1998).  it will take you
up to the level of part II Theoretical Physics and beyond. Costs less than
30 pounds, and is on discount at amazon.
Other recommended books:
Kibble and Berkshire (1985, 1996). Classical Mechanics.
Addison Wesley Longman.  This is at the same level
as the 1B course, and is good in parts.
Orderofmagnitudephysics
by Sanjoy Mahajan. [An online book.]
Classical Mechanics, Barger, VD and Olsson, MG (McGrawHill 1995)
Mechanics. Landau L D and Lifshitz E M (3^{rd} edn
ButterworthHeinemann 1976)
For Elasticity:
Lectures on Physics, Feynman R P et al. (AddisonWesley 1963) Vol 2:
Two useful chapters.
Theory of Elasticity, Landau L D & Lifshitz E M (3rd edn
ButterworthHeinemann 1995).
Mathematical Methods for Physics and Engineering, Riley RF, Hobson MP
and Bence SJ (CUP 1997).
Good history books
 On Kepler:

@book{Layzer84,
author={David Layzer},
title={Constructing the Universe},
publisher={Scientific American Library},
year={1984}
}
@book{Gingerich92,
title={ The Great Copernicus Chase : And Other Adventures in Astronomical History},
author={ Owen Gingerich },
publisher={Cambridge University Press },
year={ 1992}
}
@book{Gingerich93,
author={ Owen Gingerich },
title={The Eye of Heaven : Ptolemy, Copernicus, Kepler (Masters of Modern Physics)},
publisher={ American Institute of Physics },
year={ 1993}
}

This is the draft lecture sequence from 1999.
The actual sequence used in 1999 and 2000 can be seen by looking at the lecture notes.
There are several changes.
 The energy method.

Lagrangian and
Hamiltonian dynamics (introduced without using calculus of variations).
Illustrate with numerous examples.
 Examples of the energy method.

Compound pendulum. Rolling hoop with mass on the perimeter.
Slipping ladders. Moment of inertia. Conservation of angular momentum.
Other conserved quantities: energy, phase space volume.
Use perturbation methods to estimate error of pendulum clock.
Show Huygen's method for making isochronous pendulum and relate
to dynamics of the rolling hoop.
Particles connected to each other by strings and springs.
This leads to the central topic:
 Normal modes.

Modes of molecules.
Stability analysis using normal modes.
Square matrices as linear operators and in quadratic forms.
Eigenvectors. Transformation of linear operators and of
quadratic forms.
Perturbation theory: coupling of normal modes.
Weak coupling of nonlinear oscillators.
Taking the continuum limit of systems of masses and springs,
we come to the wave equation and elasticity.
 Elasticity.
 Definitions of strain and stress as tensors.
Young's modulus, Poisson ratio, Shear and bulk modulus.
Perturbation of wave equation on a wire by the stiffness of the wire,
for example, harmonics of piano strings.
Microscopic view of elastic behaviour. Rebound
of elastic ball or rod from hard surface.
 Central forces.

Kepler's laws.
Planetary orbits.
Perturbations of circular orbits.
Scattering. Contrast between cross sections of
hard spheres and inverse square potentials. Predicting cross sections
using dimensional analysis.
Orbits resulting from other force laws. General relativity
as a perturbation.
Gravitational slingshot.
Tides.
 Rotating frames and fictitious forces.

Centrifugal and Coriolis forces. Central force problems
reexpressed as onedimensional problems.
Nearly circular orbits revisited.
The threebody system, the Lagrange points and Trojan asteroids.
 Rigid bodies, especially the gyroscope.

Precession of gyroscope subjected to a torque. Examples:
the earth; the earthmoon system; NMR; levitron.
Free precession of a rigid body will be mentioned briefly.
Euler's equations and tennis racquet theorem if time permits.
 Interesting dynamical systems.

Possibilities include:
The driven inverted pendulum.
Different ways of driving a playground swing.
Harrison's clocks. (How to make a clock immune to linear acceleration,
centrifugal forces, and temperature variations. How to make a driving
mechanism that does not affect the period of the oscillator.)
Chaotic systems.
The relationship between a periodically driven dynamical system
and static equilibrium points of masses and springs in a periodic
potential.

Themes throughout the course
 Dimensional analysis.
Estimation.
 Conservation laws.
 Matrices. `Everything is a spring'.
 Successive approximation and perturbation expansions.
The main proposed changes compared with the 1998 course are
 Lagrangian and Hamiltonian dynamics introduced, gently.
 Normal modes are put first, rotating frames and rotating bodies later.
 Coverage of elasticity is reduced. No cantilevers.
 Coverage of rigid bodies is
reduced. Only qualitative treatment of free precession. Possibly
no Euler equations.

A small number of handouts are distributed
in lectures. Those, and my scanned lecture notes, are below.



More Physics Fun
Would you like some more fun problems?
I think you will find the following internet sites
very helpful, and fun, for thinking deeply about Physics.

 384BC322BC
 Aristotle
 310BC230BC
 Aristarchus of Samos proposed the Heliocentric theory.
 190BC120BC
 Hipparchus of Rhodes measured the angular height of the star
Alpha Virginis above the ecliptic and compared it with 150yrold
Babylonian observations. From the change of 2 degrees,
he deduced that the Earth's axis precesses at 47 arcseconds per year.
He also made detailed observations of the moon and
estimated the earthmoon distance with similar accuracy.
 150AD
 Ptolemy knocks heliocentricity on the head because
it violates Aristotle's ideas.
Building on Hipparchus's work, he
wrote a detailed mathematical theory of the motions of
the Sun, Moon, and planets.
 15641642
 Galileo
 15461601
 Tycho Brahe
 15711630
 Johannes Kepler
 16291695
 Christiaan Huygens [In 1656 he patented the first pendulum clock and applied it to longitude determination.]
 16431727 (16691687)
 Newton (Lucasian Professor in Cambridge)
 16931776
 John Harrison. Master clockmaker.

 17361813
 Lagrange. Had 10 younger siblings only one of whom survived infancy.
His father, a military man, wasted his earnings; Lagrange viewed this
as good fortune: "Had I been rich, I might never have known
Mathematics". He founded the Turin Academy of Sciences.
 1729.
 Laplace born. "The Newton of France." He published from 17661817.
Among his many achievements,
he put the (gamma1)/gamma into the speed of sound.
 1762
 Lagrange Method of Variations
 1766
 Lagrange moved to Berlin (where Euler had been).
Won prizes for work on moon, Jupiter,
3body problem and comets. Wrote "Analytical Mechanics", which
contained no diagrams.
 1775

 1787
 Lagrange moved to Paris and became depressed.
 1787
 publication commences of "Celestial Mechanics", Laplace's peak.
Biot assisted with the galleys.

 Laplace is president of board of Longitudes.
Went into politics where he was useless. Was replaced
by Bonaparte.
 1812
 Laplace's work on probability published. (Generating functions; inference)
 1889, 1892
 Poincaré. Poincaré was first to consider the possibility of chaos in a deterministic system, in his work on planetary orbits.
Little interest was shown in this work until the modern study of chaotic dynamics began in 1963.






Precession of the earth
The earth's
axis is tipped over through 23 degrees relative to the
plane of the earth's orbit round the sun (called the
ecliptic), and the orientation of the axis
relative to the stars remains
virtually constant (by conservation of angular
momentum) as the earth goes round the sun. The
equinoxes (roughly March 21 and September 21) are the
two times in the year when the earth is `sideways on'
to the sun, so that day length and night length
are equal.
The sun and the moon exert torques on the bulge, so the
angular momentum changes. As the earth's axis slowly precesses, the
time in the orbit at which the equinox occurs also
moves slowly round the sun. Hence the precession of the
earth's axis is called the
precession of the equinoxes.
The zodiacal signs correspond to 12
constellations, equally spaced along the ecliptic. The sun
does the rounds of the constellations once per year.
When the constellations were named and
identified with times of year, Aries was the constellation aligned
with the spring equinox (vernal equinox).
Since that time (3000 years ago?), the
equinoxes have precessed through a substantial angle,
so now the spring equinox occurs when the sun is
aligned with a different constellation  not
Aries, but Pisces.
However, birth signs are still allocated using
the mapping of dates to constellations that applied
3000 years ago. Since the equator is perpendicular to
the earth's axis, another way of saying where the
equinoxes are, is that the equinoxes are the intersections of
the equator and the ecliptic.
The fact that the earth precesses was known to the ancient Greeks
(get name and date), who had
sufficiently accurate historical data on the timing of the equinoxes
that they could detect the one degree per 72 years precession.

Planetariums (or planetarii?)
Events laid on for the dynamics course

Harrison's amazing chronometers
Go to Greenwich and see them still working!
Also, read the wonderful book
`The Quest for Longitude' Harvard College Press,
from which these copyright pictures are
lovingly copied.
In 1762, at the end of a 147day sea voyage, H4 (below)
had lost only 1 minute and 55 seconds.





Anecdotes
Remember the last one pound note?
Isaac Newton is on the back, and an artist has rendered
a planetary diagram next to him.
What's wrong with this picture? Answer:
They've put the sun at the centre of the ellipse!

Gravitational Slingshot
 Basics of Space Flight

N.E.A.R. trip to Eros.
Eros is one of
the largest nearEarth asteroids, with a mass thousands of times greater than
similar asteroids. "NEAR" arrived at Eros
in 1999 via a slingshot past Earth,
but failed to brake hard enough on arrival,
so didn't get into an orbit around Eros until its second attempt,
on Feb 14 2000. Since then NEAR has been in an occasionallyadjusted orbit
about Eros,
with the last close approach to Eros taking place on Oct 26 2000.

In December 2000 the Cassini spacecraft, on its journey to
Saturn, will make its close approach to the giant planet
Jupiter.
Cassini's
VenusVenusEarthJupiter Gravity Assisted journey to Saturn
includes an approach within 6 million
miles of Jupiter on December 30th.
Cassini's speed, relative to the Sun, will increase by 2.2 km/s.

Voyagers' tours of the outer planets
 Galileo's
gravity assists
with Ganymede, Callisto and Europa
(image),
and quantitative details of its assists from Venus, Earth and Earth.
More info
and VEEGA image.
More details about slingshots

Slingshot continued...
Excerpt from the above pages:
Starting out from a low Earth orbit, a spacecraft needs to increase its speed by 9 kilometers per second
(19,440 mph) in order to reach Jupiter. Navigators refer to a needed speed change as "delta V," where "delta"
indicates "change" and "V" stands for velocity.
Keep in mind, though, that Jupiter's orbit about the Sun doesn't lie in the same plane as the Earth's, so a
spacecraft going to Jupiter would have to move out of the plane of the ecliptic. This is known as a
"brokenplane" maneuver. Couldn't the spacecraft go "directly" to Jupiter without having to make the
brokenplane maneuver? Yes, but that usually means that the spacecraft needs to be going even faster to
begin with  around 11 km/sec.
By comparison, Galileo's VenusEarthEarth Gravity Assist (VEEGA) trajectory required that the
spacecraft provide a deltaV of only 4.094 km/s to reach Jupiter. Of this total, 4 km/s was provided by the IUS
booster; the other .094 km/s of deltaV came from Galileo's thrusters (the spacecraft also produced an
additional 100 meters/sec of deltaV that was used to for science purposes on the way to Jupiter, e.g. for
asteroid flybys). The additional deltaV needed to get to Jupiter was provided by the planetary flybys (2.0
km/sec (4,320 mph) from Venus, 5.2 km/sec (11,600 mph) from the first Earth flyby, 3.7 km/ sec (7,992
mph) from the second Earth flyby). Note that this doesn't add up to 9 km/sec total deltaV; that's because
we're actually giving changes in velocity (which involves direction), not just speed, and velocity changes add
as vectors.
As a bonus, Galileo didn't have to perform a brokenplane maneuver  that was thrown in "for free" by the
flybys.

The Voyager
spacecraft were an amazing feat of engineering, science and management;
they sent back more information about the solar system than
any mission before or since.
This graph shows the boosts received in the slingshots
of Voyager2 (from JPL's Basics of Space Flight).
Voyager 2 leaves Earth at about 36 km/s relative to the sun. Climbing
out, it loses much of the initial velocity the launch
vehicle provided. Nearing Jupiter, its speed is
increased by the planet's gravity, and the spacecraft's
velocity exceeds solar system escape velocity. Voyager
departs Jupiter with more sunrelative velocity than it
had on arrival. The same is seen at Saturn and
Uranus. The Neptune flyby design put Voyager close by
Neptune's moon Triton rather than attain more speed.
(From JPL's Basics of Space Flight).

Lagrange points
(From JPL's Basics of Space Flight).
Consider a system with two
large bodies, e.g. Jupiter orbiting
the sun. The third body, such as a
spacecraft or an asteroid, might
occupy any of five Lagrange points:
In line with the two large bodies are the L1, L2 and L3 points. The
leading apex of the triangle is L4; the trailing apex is L5. These last
two are also called Trojan points.
All five points are fixed points in the rotating frame.
L1, L2 and L3 are unstable fixed points, but you can stay near them
with little effort. L4 and L5 may be stable fixed points, depending
on the mass ratio of the two big bodies.

Euler equations: free precession of rigid body
Other rigid body topics mentioned in lectures
Levitation
Rolling ball on a surface
(Play these with xanim if you are on a unix system)

Typos in Hand and Finch
List of typographical errors in the course textbook,
Analytical Dynamics by Hand and Finch (C.U.P.).
Typos worth correcting
On page 9, eq 1.33, 1.37 and 1.38
all have the space between alpha and q
too small, so it looks like 1.38 for example refers to
sin^2 ( alpha q ).
In all these equations, the argument of `sin' or `cos' is just alpha
[as you can guess on dimensional grounds].
p.70, Problem 9. (Brachistrocrone)
Equation 2.77 is wrong.
It should be
y/r = arcsin[ (x/r)^{1/2} ]  [ (x/r) ( 1(x/r) ) ]^{1/2}
p.71 Problem 11. (Ski race) This question seems to me to be
illdefined. Are we to assume that the skier proceeds
at constant velocity, or that their energy is conserved?
The problem is interesting either way, neither assumption
seems plausible in real life, and I don't know which
they had in mind. I would go for the constant velocity
assumption first...
p.374
eqn (9.141) the modes should be (1,1,1), (1,0,1), and (1,2,1)
Suggestions for improvement
p.376 "energy stored in pendulum 2" should be "energy transferred
(temporarily) to pendulum 2".
p.375 "must be either odd or even" > "can be chosen to be ...."
p.373 "omega^2 = 0"  should say "this MAY correspond to a translation
or rotation".
p.372
"The Phi vectors are real numbers" > "can be chosen to be real numbers".
[There are cases, eg the three masses in a circle, where
it can be preferable for symmetry reasons to choose them
complex!]
p.364 Emphasize that the single linear transformation is
not in general orthogonal.
p.328 Figure 8.22 is confusing since the cylinder
looks symmetric.
Unimportant typos
p.417 "extra solar" > "extrasolar".
p. 142: "centrifugal force, which is repulsive, increasing as...."
SHOULD BE "decreasing as......."

Library reference numbers for the textbook
The course textbook,
Analytical Dynamics by Hand and Finch (C.U.P.),
has been acquired by some
but not all college libraries. Please harass your
college if they have not got this book.


Here are the classmarks for the text, as of 26/10/99:
[Univ. Lib.] 353:1.b.95.26 South Front, Floor 4
[Cav] 23 H 9
[Cath] 516
[Chur] 531.0151 (2 copies)
[F.C.] BCD [Hand]
[Kgs] BD Han
[Sid] BB8 D2W Han


1B Dynamics Software
My software demos are written in
gnuplot, octave and matlab.
I will put the source code here.
Unfortunately, matlab is not free software. If your college
does not have it,
(a) ask them to get it [and tell them
that 50 licenses are cheaper than 6];
(b) ask an engineer
friend to let you use theirs [all engineers use it].
 tar file of all matlab code
 Contains:
doublependulum, orbits, modes, lagrangian, rigidbody.

tar file of all gnuplot code
 contains:
bead/ beats/ liouville/ orbits/ pendulum/ strain/ (which is a matlab demo)
To run a demo, cd ; gnuplot
> load 'gnu' # (or whatever the filename is for the demo)

For supervisors only
Meet the lecturer:
Thursday October 11th 5pm Ryle Seminar Room
Thursday October 25th 5pm Ryle Seminar Room
Thursday November 8th 5pm Ryle Seminar Room
 Come and discuss your ideas for what to do in supervisions and
how to improve the lectures, questions, and solutions.
If supervisors have Any questions? then please ask.
[old html list of exercises]


A small number of handouts are distributed
in lectures. Those, and my scanned lecture notes, are below.

A small number of handouts were distributed
in lectures. My scanned lecture notes are available below.

Survey
We carried out a survey on 14th March 2000, to try
to assess the effectiveness of our physics teaching.
The results of this survey are posted here.
Many thanks to those who participated!

Part III Physics Revision Classes: Dynamics
Bryan R. Webber and David J.C. MacKay
Thanks to James Miskin and Sanjoy Mahajan
Some users of IE5 and Netscape4.7
browser have reported that they cannot
download the links on this page. The fix is, if
your browser thinks that you are at this
file .../III.html/ then you should
reload the file as
.../III.html. No slash!
Or, Try these alternative links...
Question sheet (postscript) 
pdf 
Answer to question 4 (ps)  (pdf) 
Answer to question 3 (ps)  (pdf)

Site last modified Wed Aug 17 16:39:36 BST 2005

