Lagrangian Dynamics (PHYS20401)

Prof Terry Wyatt FRS


Informal 'one-page' lecture summaries

Will appear here after each lecture.

Note: I am very grateful to Prof Mike Seymour for converting my handwritten scribbles into these more professional looking summaries.

Lecture 1: Review of Classical Mechanics in One Dimension

Lecture 2: The Lagrangian and Lagrange's Equation

Lecture 3: The Lagrangian in Plane Polar and Spherical Polar Coordinates

Lecture 4: Some Simple Examples of Applying Lagrange's Equation

Lecture 5: The Principle of Least Action

Lecture 6: The Calculus of Variations

Lecture 7: The Calculus of Variations Applied to Some Non-Mechanical Minimization Problems

Lecture 8: Two Instructive Examples of Applying Lagrange's Equation

Lecture 9: The Hamiltonian and Hamilton's Equations

Lecture 10: Some Examples of Using Hamiltonian Methods

Lecture 11: Relativity, Symmetries and Conservation Laws (part I)

Lecture 12: Relativity, Symmetries and Conservation Laws (part II)

Lecture 13: Relativity, Symmetries and Conservation Laws (part III)

Lecture 14: Relativity, Symmetries and Conservation Laws (part IV)

Lecture 15: Introduction to Coupled Oscillations and Normal Modes

Lecture 16: Formal and Informal Methods for Solutions to Equations of Motion for Coupled Oscillations

Lecture 17: Matrix Methods for Solutions to Equations of Motion for Coupled Oscillations (part I)

Lecture 18: Matrix Methods for Solutions to Equations of Motion for Coupled Oscillations (part II)

Lecture 19: Informal Lecture on Some Common Pitfalls to Avoid (part I)

Lecture 20: Informal Lecture on Some Common Pitfalls to Avoid and an Example Exam Question to Illustrate These Points (part II)

Extracurricular lectures -- not for the exam

Lecture 21: Motion of a Charged Particle in Static Electric and Magnetic Fields

Extracurricular lecture: Lagrangian dynamics for a continuous system (e.g., stretched string)


Summaries of Sections of the Course

Will appear here as we complete sections of the course

Lagrangian Methods

Hamiltonian Methods, Symmetries and Conservation Laws

Matrices: Revision Material and Exercises

Coupled Oscillations and Normal Modes using Lagrangian Methods


Example Sheets

Example sheets will be distributed in Lectures and posted here as we progress through the course. Answers will be posted here within about a week of the associated example sheet being distributed. From time to time I'll organise informal optional sessions to allow students to ask questions about the example sheets and answers.

The first two example sheets have mostly fairly simple problems to help you get used to the basic ideas and techniques of Lagrangian methods.

Example Sheet 1 ...... Answers.

Example Sheet 2 ...... Answers.

Sheet 3 has some problems to help you practice using the calculus of variations.

Example Sheet 3 ...... Answers.

Sheet 4 has some simple problems to help you get the hang of analysing systems using Hamiltonian techniques. It also has some more examples for Lagrangian methods; some of these are at about the level of complexity of those considered in recent exams.

Example Sheet 4 ...... Answers.

Sheet 5 has some simple problems on Poisson brackets, generators of transformations, symmetries and conservation laws.

Example Sheet 5 ...... Answers.

Sheet 6 has some problems on normal mode oscillations.

Example Sheet 6 ...... Answers.

Sheet 7 has some more problems on normal mode oscillations.

Example Sheet 7 ...... Answers.

Sheet 8 has revision problems on many aspects of the course.

Example Sheet 8 ...... Answers: Question 1 .... Question 2 .... Question 3 .... Question 4

I hope that you try your best to solve the problems (and think of sensible ways of checking your answers) before you look at my answers!

If even after looking at the provided answers you can't see how you went wrong, or you'd like feedback on any other aspect of your answers, please feel free to: come along to the optional Lagrangian sessions I'll organise from time to time, or ask me after a lecture, or come and ask me in person in Room 6.22 (Schuster), or send me email.

Many other relevant problems can be found on the web, for example under the header "Links to other written material" below.

Optional Examples Classes

15th November: optional examples class 1 - Problems ...... Answers

22nd November: optional examples class 2 - Problems ...... Answers

22nd November: optional examples class 3 - Problems ...... Answers

13th December: optional examples class 4 - Problems ...... Answers


Past Exams: Papers, Bottom Line Answers and Feedback

Here is material relating to the exams since I started teaching the course in the academic year 2010-2011:

2016-2017 (Course given by Prof. Mike Seymour): Exam ...... Bottom Line Answers ...... Feedback

2015-2016: Exam ...... Bottom Line Answers ...... Feedback

2014-2015: Exam ...... Bottom Line Answers ...... Feedback

2013-2014: Exam ...... Bottom Line Answers

2012-2013: Exam ...... Bottom Line Answers ...... Feedback

2011-2012: Exam ...... Bottom Line Answers ...... Feedback

2010-2011: Exam ...... Bottom Line Answers ...... Feedback

Some material relating to exams before I took over the course may be found on Blackboard. However, I am afraid I cannot vouch for the completeness of the material provided by previous lecturers.


Other Supplementary Material

The syllabus for the course as given in the Blue Book

Lecture 5: Illustration of the Principle of Least Action for V = mgx


Recommended Textbooks:

Kibble, T.W.B. & Berkshire, F.H. "Classical Mechanics", 5th edition (Longman). This is a good, basic book on Classical Mechanics. It covers Lagrangian and Hamiltonian mechanics at about the level of this course, in addition to material that would be useful revision from the first year courses "Dynamics" (PHYS 10101) and "Advanced Dynamics" (PHYS 10672). The Calculus of Variations and the Principle of Least Action are introduced in Section 3.5-3.7. However, much of the book uses the "Newton's Laws" approach. It is not until Chapter 10 onwards that Lagrangian methods are used as the default approach. Chapters 10-12 present the material most relevant to this course. There are useful problems and one-line answers are provided.

Goldstein, H., Poole, C. & Safko, J. "Classical Mechanics", 3rd edition (Addison-Wesley). This is a much more substantial, advanced and mathematically rigorous book than the book by Kibble. Lagrangian methods are used throughout. Chapters 1, 2, 6 and 8 are the ones most relevant to this course. I strongly recommend a serious and thorough study of this book to anyone who is planning on taking the advanced theoretical courses that follow on from this one. There are lots of useful problems, but no answers are provided.

Also worth reading is Landau, L.D & Lifshitz, E.M. "Mechanics", 3rd Edition (Pergamon Press). This is a very concise and elegant treatment by one of the greatest theoretical physicists of the 20th century. (In 160 pages it covers in essence the material that takes Goldstein 600 pages!) Lagrangian methods are used throughout. Chapters 1, 2, 5 and 7 are the ones most relevant to this course. There are useful problems and and brief worked solutions are provided. (Note: the definition of the Poisson Bracket in page 135 of this book has a relative minus sign compared to the definition used in the lectures and in most other books.)

For an informal introduction to The Principle of Least Action by one of my (and perhaps your) heroes you might like to try reading: Chapter 19, Volume 2 of the Feynman Lectures on Physics, "The Principle of Least Action".


Demonstrations, Applets, etc.

There are lots of interesting and/or interactive demonstrations, applets, etc., out there on the web. Here are pointers to a few. If you come across any others that you find particularly illuminating, interesting, entertaining, please do let me know so that I can add a link to it here! N.B. For some of these you'll need to have Java installed on your computer and enabled in your browser. For the Wolfram Demonstrations Project pages you may need to install the Wolfram CDF player.

Applet demonstrating coupled oscillators (as used in Lecture 15)

Interactive principle of least action.

Demonstration of a ball rolling in an inverted cone. It's interesting to compare the motion of a solid and hollow ball of the same mass and radius. N.B. To simulate the simpler system of a point-like mass sliding inside an inverted cone, as discussed in Lecture 8, click on "Ball Type: Other" and set the parameter "Ball Inertia" to be 0.0.

There are lots of interesting demonstrations at the Wolfram Demonstrations Project. Take a look, e.g., at ...... Bead sliding on a rotating circular wire ...... Ball Rolling without Slipping inside a Vertical Cylinder (Demo 1) ...... Ball Rolling without Slipping inside a Vertical Cylinder (Demo 2) ...... Euler Angles ...... Double Pendulum ...... ...... ...... ......


Links to other written material relevant to Lagrangian Dynamics, etc.

There are lots of web-sites for courses on Lagrangian Dynamics, sets of lecture notes, etc., out there on the web. Here are pointers to a few. Of course, none of these are aimed at exactly the same level or have exactly the same syllabus as PHYS20401. However, reading around the syllabus will hopefully make attending PHYS20401 (even) more fun! If you come across any other material on the web that you find particularly illuminating, interesting, entertaining, please do let me know so that I can add a link to it here!

David Tong: Lectures on Classical Dynamics

David MakKay: Dynamics Lectures

James Binney: Classical Mechanics

Edwin Taylor: When action is not least - a discussion of when the physical path can correspond to the action not being a minimum.

Daniel Arovas: Calculus of Variations ...... Lagrangian Dynamics ...... Noether's Theorem ...... Constraints ...... Normal Modes

Richard Fitzpatrick: Classical Mechanics

The John Ryland's library has a number of useful and relevant books available to download electronically. You can search the catalogue online.


Some video recordings of interesting lectures

Richard Feynman: The Character of Physical Law - I recommend particularly lectures 1-4 for their relevance to this course.


Just for fun ;-)

Brian and Harry discuss the concept of time

Who invented the theory of relativity? (QI version)


Your Questions, Feedback, etc

Do you have any questions about the physics or example sheets? Do you have any feedback on the lectures or other aspects of the course? Did you find any mistakes anywhere? I'd be very interested to hear from you! Please come and find me in Room 6.22 (Schuster) or:

Send email to userid: twyatt at the mailserver: fnal.gov