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1B Dynamics:
16 Lectures by David J.C. MacKay
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. EndPage
Part III Physics Revision Classes: Dynamics
1B Dynamics: Synopsis
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: Lagrangian and Hamiltonian dynamics. State-space 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. Transformation of linear operators and of quadratic forms. Perturbation theory: coupling of normal modes. Weak coupling of nonlinear oscillators. 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. Perturbations of circular orbits. Scattering. Cross sections of hard spheres and inverse square potentials. Predicting cross sections using dimensional analysis. Orbits resulting from other force laws. Orbital transfers. Gravitational slingshot. Tides.
Rotating frames and fictitious forces: Centrifugal and Coriolis forces. Central force problems re-expressed as one-dimensional problems. The three-body system, the Lagrange points and Trojan asteroids.
Rigid bodies: Relationship between angular velocity vector and angular momentum vector. Precession of gyroscope subjected to a torque. Examples: the Earth; the Earth-Moon system; NMR; levitron. Free precession of a rigid body.
Applications of dynamical systems: The driven inverted pendulum. Different ways of driving a playground swing. Harrison's clocks. Chaotic systems.
EndPage Topic course;book Alias books.html Page BooksBOOKS - can be bought from Heffers, CUP bookshop, or http://www.amazon.co.uk/stu-text
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.
Order-of-magnitude-physics by Sanjoy Mahajan. [An online book.]
Classical Mechanics, Barger, VD and Olsson, MG (McGraw-Hill 1995)
Mechanics. Landau L D and Lifshitz E M (3rd edn Butterworth-Heinemann 1976)
For Elasticity:
Lectures on Physics, Feynman R P et al. (Addison-Wesley 1963) Vol 2: Two useful chapters.
Theory of Elasticity, Landau L D & Lifshitz E M (3rd edn Butterworth-Heinemann 1995).
Mathematical Methods for Physics and Engineering, Riley RF, Hobson MP and Bence SJ (CUP 1997).
EndPage Alias sequence.html Page Lecture sequenceProposed lecture sequence
This is the draft lecture sequence from 1999. The actual sequence used in 1999 can be seen by looking at the lecture notes.- 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
re-expressed as one-dimensional problems.
Nearly circular orbits revisited.
The three-body system, the Lagrange points and Trojan asteroids.
- Rigid bodies, especially the gyroscope.
-
Precession of gyroscope subjected to a torque. Examples:
the earth; the earth-moon 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.
Comments
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.
EndPage Alias notes.html Topic course;handouts SectionPage Lecture notes
Lecture notes for 2000
A small number of handouts are distributed in lectures. Those, and my scanned lecture notes, are below.Exercises
-
Problem sheet (postscript).
For corrections to the version handed out in October 2000, please
go to \metafaq.
- The old 1999 Problem sheet (postscript).
- The ancient 1998 Problem sheet (postscript).
Worked Solutions1B students: you're encouraged not to look at these solutions before you have worked hard on them yourselves.Worked solutions to some traditional exercises from 2000 (postscript) | (pdf) Worked solutions to quickies (some from 1999) Worked solutions to traditional exercises from 1999 (scanned images) Worked solutions to deep thought from 1999
|
Links related to exercises
- Liquid lengths (problem Q2.)
- Kater's pendulum: experimental guide from 1972 (problem T2.)
- Maximum walking speed in reduced gravity
- Equation of time (Daylength) |
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.-
Yacov Kantor's Physics QUIZ
tends to have quite tough questions, usually about theoretical
thought experiments. (A few of them still haven't been solved!)
- `Physics question of the week'
from UMD usually has easier problems, all of which
relate to real physics demonstrations; each question and answer
are accompanied by pictures or movies showing the experiment being
done.
You may find my page of
comments on these questions is
a helpful way to get into them, because I've included an index and
marked my favourite questions.
- Probably the best way of all to learn Physics is to teach it to someone else. You may therefore find it helpful to dig around among the other Physics teaching resources on the net.
-
Order-of-magnitude-physics by Sanjoy Mahajan. [An online book.]
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.
EndPage Alias planetary.html Topic course;software Page Planetary dynamicsPlanetariums (or planetaria?)
- The Nine Planets | Xephem planetarium software | Realtime picture of Jupiter and moons | Web-based Sky map - Map now |
- Planetarium images |
- Space station sightings | Space station now |
Events laid on for the dynamics course
- Leonid meteor storm - November 14-21, sadly not expected to be as good as previous years; | More about Leonids | and more Meteor showers |
Gravitational Slingshot
-
N.E.A.R. trip to Eros.
Eros is one of
the largest near-Earth 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 occasionally-adjusted 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 Venus-Venus-Earth-Jupiter 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.
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 "broken-plane" maneuver. Couldn't the spacecraft go "directly" to Jupiter without having to make the broken-plane 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 Venus-Earth-Earth Gravity Assist (VEEGA) trajectory required that the spacecraft provide a delta-V 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 delta-V came from Galileo's thrusters (the spacecraft also produced an additional 100 meters/sec of delta-V that was used to for science purposes on the way to Jupiter, e.g. for asteroid flybys). The additional delta-V 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 delta-V; 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 broken-plane maneuver -- that was thrown in "for free" by the flybys.
Euler equations: free precession of rigid body
Other rigid body topics mentioned in lectures
Levitation
EndPage Alias admin.html Topic course Section Administrative stuff Alias typos.html Topic course;book Page Typos in the textbookTypos 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 ill-defined. 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" -> "extra-solar".
p. 142: "centrifugal force, which is repulsive, increasing as...." SHOULD BE "decreasing as......."
EndPage Alias libraries.html Topic course;book Page Finding the textbookLibrary 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.
- 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)
EndPage Alias supervisors.html Topic course;supervisors SectionPage For supervisors
EndPage Alias notes99.html Topic course;handouts SectionPage 1999 Lecture notesFor supervisors only
There's nothing special for supervisors at the moment, but if supervisors have \metafaq then please ask.
[old html list of exercises]Lecture notes from 1999
A small number of handouts were distributed in lectures. My scanned lecture notes are available below.Part III Physics Revision Classes: Dynamics
Bryan R. Webber and David J.C. MacKay
1999-2000 (These questions have been used as the revision questions for several years.) 2000-2001 Question sheet (postscript)
Question sheet (pdf)
Question sheet (html) [This html page is broken, only look at it as a last resort.]Answers to questions 1 and 2 (scanned images).
Answer to question 3 (postscript)
Answer to question 3 (pdf)
Answer to question 4 (postscript)
Answer to question 4 (pdf)
Question sheet (postscript)
Question sheet (pdf)
Answers to questions 1 and 2 (scanned images).
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)
EndPage Alias survey.html Topic course;survey Page Teaching SurveySurvey
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! EndPage
