Physics 3210

Classical Mechanics and Mathematical Methods 2

Spring 2008





Fun Stuff:


            Jeff Dunham and Walter #1                                        Fifty Years of Teaching Math

            Jeff Dunham and Walter #2                                        ItÕs Hard to Kiss the Lips É

            Jeff Dunham and Walter #3                                        Five House Puzzle     

            Achmed the Dead Terrorist                                         Ship and Boiler

            YMCA Puppets                                                          Beer Can Tosser



Lectures: MWF 2:00 – 2:50 in Duane G-2B47

Help Sessions: Mondays at 5:00 in Duane G-2B21.


Instructor: Joel Broida


Offices:  Gamow Tower F919      tel: 303-492-2011

               ECCR 251                      tel: 303-492-4543

Home Phone (usually the best place to find me): 303-993-7220

Office Hours: After class and by appointment


Graders: Jeremy Nuger

                 Office: Duane F929

                 Help Room hours: Monday 2 – 3, Tuesday 12 – 1, Wednesday 2 – 3.


                 Craig Hogle

                 Office: Duane C119


Grading: Homework 60%, midterm and final each 20%.


Midterm: 7:00 pm Thursday, March 13 in the classroom.

Final Exam: 1:30 pm Saturday, May 3 in the classroom.


Texts: The official texts are:   Thornton and Marion Classical Dynamics of Particles and Systems

                                                Taylor Classical Mechanics

                                                Boas Mathematical Methods in the Physical Sciences


These and some of the ones mentioned below are on reserve in Lester Library. For the most part I wonÕt be following any of these books in detail, but rather I will use them for additional references and examples. Since I donÕt think very many students can take notes and still listen to the lectures carefully at the same time, I will be handing out detailed (mostly handwritten) lecture notes so you can pay attention and ask meaningful questions. Of course, if attendance goes down the notes will stop. (What did you expect?) Or maybe the homework will just get harder since youÕll have more time to work on it if you arenÕt in class.


Older editions of many books are a lot better than the newer ones that were written by other people. If necessary, and if you can find them online, other books you might find useful are: (1) the second edition of MarionÕs Classical Dynamics of Particles and Systems; (2) the third edition of SymonÕs Classical Mechanics; (3) the second edition of GoldsteinÕs Classical Mechanics. Two other older books that are somewhat more advanced than this course but are excellent for what they cover are: (4) Fetter and Walecka Theoretical Mechanics of Particles and Continua; (5) Byron and Fuller Mathematics of Classical and Quantum Physics. Both of these are available as Dover paperbacks, so they are very inexpensive. You might also find some of the SchaumÕs Outlines useful in a number of subjects.


Overview of Course:


These are the Contents/Index for the various topics we will cover. The links will simply open for you to peruse.


            Section 1 Contents – Linear Algebra part 1

            Section 2 Contents – Calculus of Variations and LagrangeÕs Equations

            Section 3 Contents – Central Force Motion

            Section 4 Contents – Motion of a System of Particles

            Section 5 Contents – Linear Algebra Part 2

            Section 6 Contents – Coupled Oscillations


Rough Schedule of Time per Section:   Section 1 – 4 weeks

                                                                  Section 2 – 3 weeks

                                                                  Section 3 – 1 week

                                                                  Section 4 – 3 weeks

                                                                  Section 5 – 3 weeks

                                                                  Section 6 – 1 week


Of course, this is only a very approximate goal. We can spend either more or less time on any section depending on how the lectures are progressing.


Lecture Notes:


Here are the scanned lecture notes in pdf format. More will be posted as we get closer to needing them. The larger sections are broken into smaller files that I call chapters, but itÕs a very informal distinction. I strongly recommend that you print or somehow have the notes ahead of time, and bring them to class with you. We also may or may not be able to cover all of these notes in class, but they are all here for completeness.


The lecture notes are available at the campus Printing Services office located on the east side of the stadium. Go in Gate 11 and up to the second floor. You can pay them by cash, check or credit card.


The notes for all of Section 1 will be available by 2:00 Thursday, January 17. The cost is approximately $10.


            Section 1 – Linear Algebra Part 1

                                Chapter 1 – Vector Spaces

                                Chapter 2 – Inner product spaces

                                Chapter 3 – Linear Equations and Matrices

                                Chapter 4 – Determinants

                                Chapter 5 – Linear Transformations

                                Chapter 5 pages 140-141 replaced by these pages 140-143



The notes for Section 2 will be available by Wednesday, February 6. The cost should be around $7.


            Section 2 – Calculus of Variations and LagrangeÕs Equations

                                Chapter 1 – Calculus of Variations

                                Chapter 2 – DÕAlembertÕs Principle and LagrangeÕs Equations

                                Chapter 3 – HamiltonÕs Equations and Vector Calculus

                                Supplementary pages 42a-b to Chapter 2.



The notes for Sections 3 and 4 should be available Tuesday, March 4. The cost should be around $6.


            Section 3 – Central Force Motion

                                Central Forces and KeplerÕs Laws


            Section 4 – Motion of a System of Particles

                                Chapter 1 – Momentum, Angular Momentum and Torque

                                Chapter 2 – The Inertia Tensor

                                Chapter 3 – EulerÕs Equations of Motion



The notes for Sections 5 and 6 will be available Monday afternoon, April 7. The cost should be around $9.


            Section 5 – Linear Algebra Part 2

                                Chapter 1 – Eigenvalues and Eigenvectors

                                Chapter 2 – Invariant Subspaces and Geometric Multiplicity

                                Chapter 3 – Diagonalization of Normal Matrices

                                Chapter 4 – Diagonalization of Quadratic Forms

                                Chapter 5 – Overview of Diagonalization

                                Chapter 6 – The Adjoint Operator


            Section 6 – Coupled Oscillations

                                Coupled Oscillations


This course is for YOUR benefit. If there are specific topics that you would like to cover, please let me know and IÕll do the best I can within the time constraints we have and how much work you are willing to do.


Reading Assignments:


Please read the pages of notes indicated before the listed lecture date. I will assume that you have read this material, and in the lecture I will concentrate on the more difficult portions of the chapter. Of course, if you have any questions dealing with the reading assignment, please ask in class.


      Linear Algebra Part 1

      Date             Pages                       Date             Pages                       Date             Pages

                                                            Jan 23           40 – 51                     Jan 25           57 – 62

      Jan 28           72 – 76                     Jan 30           77 – 83                     Feb 1            90 – 93

      Feb 4            104 – 109                 Feb 6            115 – 119                 Feb 8            125 – 132

      Feb 11          133 – 141                


      Calculus of Variations and LagrangeÕs Equations

      Date             Pages                       Date             Pages                       Date             Pages

                                                            Feb 13          1 – 4                         Feb 15          5 – 13

      Feb 18          11 – 16                     Feb 20          17 – 21                     Feb 22          22 – 28

      Feb 25          29 – 35                     Feb 27          39 – 44                     Feb 29          48 – 54

      Mar 3           55 – 68                    

(Also see Boas, Chapter 9; Thornton & Marion, Chapters 6 and 7; Taylor, Chapters 6, 7 and 13.)


      Central Force Motion

      Date             Pages                       Date             Pages                       Date             Pages

                                                            Mar 5           1 – 8                         Mar 7           9 – 17

      Mar 10         18 – 25                    

(Also see Thornton & Marion, Chapter 8; and Taylor, Chapter 8.)


      Motion of a System of Particles

      Date             Pages                       Date             Pages                       Date             Pages

                                                            Mar 12         1 – 6d                       Mar 14         7 – 10

      Mar 17         11 – 16b                   Mar 19         17 – 21                     Mar 21         22 – 29

      Mar 31         30 – 32                     Apr 2            33 – 40                     Apr 4            41 – 46

(Also see Thornton & Marion, Chapters 9 and 11; Taylor, Chapters 3, 4 and 10.)


      Linear Algebra Part 2

      Date             Pages                       Date             Pages                       Date             Pages

      April 7          1 – 11                       April 9          22 – 30                     April 11        36 – 39

      April 14        47 – 53a                   April 16        74 - 84

(For a basic treatment of diagonalization, you might also want to look at various sections in the book Linear Algebra: A Modern Introduction by David Poole. ItÕs on reserve in Lester Library.)


      Coupled Oscillations

      Date             Pages                       Date             Pages                       Date             Pages

                                                                                                                  April 18        1 – 6

      April 21        7 – 20                       April 23        21 - 29

(Also see Thornton & Marion, Chapter 12; Taylor, Chapter 11 and his Appendix.)




There will probably be one problem set due each Wednesday afternoon that covers the previous weeks work. Since neither the graders nor I are mind readers, be sure to show all your work legibly and in a logical manner that we can follow. Do your scratch calculations on scratch paper, and hand in only a finished presentation. Writing out the solutions as if you were explaining them to someone not only helps with the grading, it will also help you to make sure that you really understand what you are doing.


            Problem Set 1 – due January 23. Covers LA Part 1 notes through page 39.

            Problem Set 2 – due January 30. Covers LA Part 1 notes through page 71.

            Problem Set 3 – due February 6. Covers LA Part 1 notes through page 103.

            Problem Set 4 – due February 13. Covers LA Part 1 notes through page 141 (the end).

            Problem Set 5 – due February 25. Covers Calc of Var notes through page 28.

            Problem Set 6 – due February 29. Covers Calc of Var notes through page 42b.

            Problem Set 7 – due March 7. Covers Calc of Var notes through page 68.

            Problem Set 8 – due March 17. Covers Central Force Motion notes.

            Problem Set 9 – due March 21. Covers Systems of Particles notes through page 18.

            Problem Set 10 – due April 9. Covers Systems of Particles notes through page 40.

            Problem Set 11 – due April 16. Covers Systems of Particles notes through page 56 and LA Part 2 notes through page 20.

            Problem Set 12 – due April 23. Covers LA Part 2 notes through page 84.

            Problem Set 13 – due April 30. Covers Coupled Oscillations notes.


Homework Solutions: These will be posted after the homework sets have been turned in. Keep in mind that these are how I did the problems, and there are frequently other equally valid approaches.


            PS1 Solutions                         PS2 Solutions                         PS3 Solutions

            PS4 Solutions                         PS5 Solutions                         PS6 Solutions

            PS7 Solutions                         PS8 Solutions                         PS9 Solutions

            PS10 Solutions                       PS11 Solutions                       PS12 Solutions

            PS13 Solutions


Midterm Exam and Solutions Here are the Midterm and its Solutions.


Final Exam and Solutions Here is the Final and its Solutions.


Some Background Math Notes:


Here are some sections from ThomasÕs Calculus on Lagrange multipliers and TaylorÕs formula in two variables.


Lagrange Multipliers               TaylorÕs Formula


And here are some more excellent chapters from a couple of advanced calculus books on differentiation in R^n (and TaylorÕs theorem).


Differential Calculus               Differentiability on Rn


Here is the chapter on complex numbers and hyperbolic functions from the book Mathematical Methods for Physics and Engineering by Riley, Hobson & Bence.


Complex Numbers


And here are some appendices from my book that also cover a lot of basic mathematical concepts that you should be familiar with.


Appendix A – Sets, mappings, induction and complex numbers (as well as some other topics).

Appendix D – Metric spaces, open and closed sets, compactness and the Fundamental Theorem of Algebra.

Appendix E – Sequences and series of numbers, and some basic topology.

Appendix F – A brief discussion of path connectedness (necessary for a treatment of oriented vector spaces).



Miscellaneous Notes:


This first set of notes is on the Dirac delta function and a famous theorem of Weierstrass that is the basis for Fourier series and the expansion by other families of orthogonal polynomials (like the Legendre polynomials). These notes are a bit more mathematically demanding than most of what we will be covering in class, but donÕt worry. We wonÕt be covering this in class at all (unless there is some real interest on your part), so the notes are here solely for your benefit. Virtually all QM books give a very imprecise introduction to the delta function, so this should give you a good background to make sense out of what you will see elsewhere. (Weierstrass ÒinventedÓ the delta function about 35 years before Dirac, and did it fairly rigorously. This is not to take anything away from DiracÕs brilliance in inventing it himself for his own purposes in formulating quantum mechanics.)


            Dirac delta function


Here are some other miscellaneous notes that I wrote for one reason or another when I was teaching myself LaTeX. You should find them useful in other classes. We wonÕt be covering any of them in this course. (Again, unless you want to.)


            Lorentz transformation

            Gamma Function, StirlingÕs Formula and volume of an n-sphere

            Contour integral example



Some Biographical Sketches (from the outstanding book Differential Equations with Applications and Historical Notes by George Simmons)


            Gauss                          Euler                            Riemann



And I end with a quote from the recent paper Mathematics as the language of physics (


ÒMathematics is undoubtedly a subject which deserves to be studied in its own right. As such it provides a superb exercise for the mind and can easily be seen to lead to people possessed of a flexible thinking which may be applied in myriad areas. However, mathematics also has a vital role to play as the language of physics. As such, the part it plays must be subservient to the physics but, having said that, its role is no less important than that it enjoys when it is studied as a subject in its own right. In both roles the conditions affecting the validity of results are equally important and it is unfortunate that this aspect of mathematics – so important to the pure mathematician – should appear to be overlooked on so many occasions within physics.Ó


J. Dunning-Davies

Department of Physics

University of Hull, England