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1B Dynamics:

16 Lectures by David J.C. MacKay

From 1999 to 2001 inclusive I taught the 2nd-year 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.

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: Equations of motion from energy functions. Central force problems re-expressed as one-dimensional problems. Perturbations of circular orbits. 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. 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.

BOOKS - can be bought from Heffers, CUP bookshop, or

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).

Good history books

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

Proposed lecture sequence

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.

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.


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 on how the 1999 course differed from what went before

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

  1. Lagrangian and Hamiltonian dynamics introduced, gently.
  2. Normal modes are put first, rotating frames and rotating bodies later.
  3. Coverage of elasticity is reduced. No cantilevers.
  4. Coverage of rigid bodies is reduced. Only qualitative treatment of free precession. Possibly no Euler equations.

Lecture notes for 2001

A small number of handouts are distributed in lectures. Those, and my scanned lecture notes, are below.
lecture notes handouts and other useful stuff
Lecture 1: The Energy Method.
Lecture 2: Dimensional Analysis Handout 2: Dimensional analysis (html) / (postscript) | (pdf) |
Lecture 3: Newtonian Dynamics Revision Handout 3: Newtonian Dynamics Revision (postscript) | (pdf) |
Lecture 3-4: Extensions of the energy method Planetary assignment
Handout 4: Almost Inverse-square force-law orbits (postscript) | pdf | old Handwritten version |
Lecture 5: Perturbation Expansions, State space diagrams + Lagrangian Dynamics. State space diagram of pendulum
Lecture 6: Perturbation expansions + Lagrangian Dynamics, continued.
Lecture 7: Pendulums, Lagrangian Dynamics, and the Hamiltonian.
Lecture 8: Hamiltonian Dynamics. Bead dynamics from the Hamiltonian
Handout 5: Hamiltonian dynamics summary and example (postscript) | pdf | Liouville's theorem for vertically bouncing balls |
Lecture 9: Normal Modes I . Handout 6: Drag force by dimensional analysis (postscript) | pdf |
About surface tension and latent heat
Molecule modes: Quicktime Movie by Luca Turin (more info)
Lecture 10: Normal Modes II .
Handouts 7a & 7: Normal modes - 7a (postscript, 2 pages) | pdf | 7 (postscript, 8 pages) | pdf |
Lecture 11: Normal Modes III .
Lecture 12: Normal Modes IV . Symmetries and eigenvectors - the complex wave (corkscrew) is an eigenvector of the translation operator
Lecture 13: Elasticity . | Beats |
Handout 8 contains the following sections:
1: | Strain and stress - postscript | | pdf |
2: | Kepler's Ellipses, and gravitational slingshots (postscript) | pdf | orbit diagram |
3: | Rotating frames (postscript) | pdf |
4: | Optional reading - Inverted pendulum | pdf |
Lecture 14: Kepler and orbits . | Marimba | History of planetary observations | Orbits in inverse-square potential |
Lecture 15: Fictitious forces in rotating frames .
Lecture 16: Rigid bodies . (A subset of these notes were used in the 2001 lecture 16)


If you would like to find more fun Physics problems, please check out the Physics fun link.

Links related to exercises

Worked Solutions

1B students: you're encouraged not to look at these solutions before you have worked hard on them yourselves.

Up to date solutions

Solutions, 2001 (postscript) | pdf.
Solution to the precession of the earth problem (T31) by Gareth Rees (wgr2)

Solutions to past exam questions

Somewhat outdated solutions that may fill in gaps in the up to date solutions

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

Warm fuzzy feeling awards

Thankyou to the following generous people for help with scanning in 1999:
Iain Murray
Bob Butcher
James Ransley

H.T. Leung's worked solutions to 1999 question sheet (compressed postscript, 1Megabyte) or pdf 1M

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.

A Brief History of Dynamics (see recommendations in the reading list)

Aristarchus of Samos proposed the Heliocentric theory.
Hipparchus of Rhodes measured the angular height of the star Alpha Virginis above the ecliptic and compared it with 150-yr-old Babylonian observations. From the change of 2 degrees, he deduced that the Earth's axis precesses at 47 arc-seconds per year. He also made detailed observations of the moon and estimated the earth-moon distance with similar accuracy.
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.
Tycho Brahe
Johannes Kepler
Christiaan Huygens [In 1656 he patented the first pendulum clock and applied it to longitude determination.]
1643-1727 (1669-1687)
Newton (Lucasian Professor in Cambridge)
John Harrison. Master clockmaker.
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.
Laplace born. "The Newton of France." He published from 1766-1817. Among his many achievements, he put the (gamma-1)/gamma into the speed of sound.
Lagrange Method of Variations
Lagrange moved to Berlin (where Euler had been). Won prizes for work on moon, Jupiter, 3-body problem and comets. Wrote "Analytical Mechanics", which contained no diagrams.
Nevil Maskelyne and Schiehallion, famous gravitating mountain, now owned in part by the John Muir Trust (Help to buy it!)
Lagrange moved to Paris and became depressed.
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.
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.

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 147-day sea voyage, H4 (below) had lost only 1 minute and 55 seconds.

H1 (c) Harvard College Press
H4 (c) Harvard College Press Harrison (c) Harvard College Press



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

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 "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.


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).

images/bsf16-22voyager 2

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 sun-relative 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

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

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 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......."

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]

Lecture notes for 2000

A small number of handouts are distributed in lectures. Those, and my scanned lecture notes, are below.
lecture notes handouts and other useful stuff
Lecture 1: The Energy Method.
Lecture 2: Dimensional Analysis Further reading about dimensional analysis (html) / (postscript) |
Lecture 3-4: Extensions of the energy method Planetary dynamics observations
Lecture 4: Perturbation Expansions .
Lecture 5: State space diagrams, Lagrangian Dynamics. State space diagram of pendulum
Handout 2: Almost Inverse-square force-law orbits (postscript) | pdf | old Handwritten version |
Handout 3: Wonky pendulum solution (postscript) | pdf |
Lecture 6: Lagrangian Dynamics, continued; the Hamiltonian. Kater's pendulum
Lecture 7: Hamiltonian Dynamics. Bead dynamics from the Hamiltonian
Handout 4: Drag force by dimensional analysis (postscript) | pdf |
Handout 5: Hamiltonian dynamics summary and example (postscript) | pdf |
Lecture 8: Normal Modes I . Molecule modes: Quicktime Movie by Luca Turin (more info)
Lecture 9: Normal Modes II . Liouville's theorem for vertically bouncing balls Kater's pendulum, further information
Lecture 10: Normal Modes III . Handout 6: Normal modes (postscript, 8 pages) | pdf |
Lecture 11: Normal Modes IV . Beats
Lecture 12: Normal Modes V . Marimba
Lecture 13: Elasticity . | H7: Strain and stress - postscript | | pdf |
Lecture 14: Kepler and orbits . History of planetary observations | Orbits in inverse-square potential | H8: Kepler's Ellipses, and gravitational slingshots (postscript) | pdf |
Lecture 15: Fictitious forces in rotating frames . H9: Rotating frames (postscript) | pdf |
Lecture 16: Rigid bodies . Optional reading - Inverted pendulum |

Lecture notes from 1999

A small number of handouts were distributed in lectures. My scanned lecture notes are available below.
Lecture 1. | Course synopsis. | Handout 2 (Reading recommendations and rough lecture plan).
Lecture 2. | H3: Numbers suitable for use on backs of envelopes (html) / (postscript) | H4: About dimensional analysis (html) / (postscript) |
Lecture 3 | H5: Almost Inverse-square force-law orbits |
Lecture 4. | H6: Planet-gazing |
Lecture 5.
Lecture 6 | H7: Summation convention |
Lecture 7.
Lecture 8. | H8a: Eigenvectors of translation-invariant systems (postscript) | H8b: Cross-products (postscript) |
Lecture 9. | H9: Strain and stress |
Lecture 10. | Planetarium images | H10: Kepler's Ellipses |
Lecture 11.
Lecture 12. | H11: Slingshot |
Lecture 13.
Lecture 14.
Lecture 15. | Rigid body images |
Lecture 16. | H12: Details of L16 calculations | Optional reading - H13: Inverted pendulum |


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

2000-2004 1999-2000 (These questions were used as the revision questions for several years.)
Question sheet (postscript)
Question sheet (pdf)

Answers to question 1 (scanned images).

Qu 2 (bike)

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)
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)

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