Physics 106ab
Fall Quarter, 2006 and Winter Quarter, 2007
Course Homepage


The Ph106abc sequence covers classical mechanics and electromagnetism at a level of sophistication beyond the introductory Ph1/2/12 sequence.  You will learn a variety of new techniques and formalism that will allow you to attack a wider set of problems than you saw in the introductory sequences as well as provide you a deeper, more unified understanding of the structure and fundamental principles of classical physics.  Ph106a and the first half of Ph106b will cover classical mechanics and special relativity.  The remainder of Ph106b and all of Ph106c will cover classical electromagnetism.

This is the home page for the classical mechanics/special relativity section of Ph106ab.

The classical electromagnetism half of Ph106bc will be taught by Prof. Cross.

Quick Links


archive below

Vital Information

Location: 107 Downs
Time: TuTh 10:30 am - 12:00 pm
Prof. Sunil Golwala, 311 Downs, Mail Code 59-33, golwala at
Teaching Assistants:
Andrey Rodionov, rodionov at
Peng Wang, pengw at
Please contact the TAs directly if you would like to make appointments outside of normal office hours.

Office Hours and Contact Information:

Prof. Golwala:  9:00 pm-11:00 pm Thursdays, 311 Downs (may move to 107 Downs if attendance is high).  Additional office hours by appointment or by popular demand.  If you need to contact me outside of office hours, please try email first.  I am happy to arrange meetings outside of normal office hours, but I am not always available on the spur of the moment.  Please include "Ph106" in the subject line of your email -- I get a lot of email, and I want to make sure I see your emails quickly.

Wednesdays, 9:00 pm - 11:00 pm Sherman Fairchild Library Room 231, on the 2nd floor
The TAs will alternate on the Wednesday office hours.

Feedback: I greatly appreciate student feedback; feedback prior to the end-of-term evaluations lets me modify the class to fit your needs.  In person, by email, by campus mail, whatever you like.  If you would like to preserve your anonymity, campus mail will usually work.  I have mailboxes on the 3rd floor of Downs near my office and in 61 W. Bridge. 

Ombudspersons: I would also like to have two student ombudspersons for the class.  Contact me to volunteer.


Lecture Strategy and In-Class Problems

I will not cover in lecture every bit of material you will be responsible for.  There are some topics that are really better covered by reading than by lecture, and some topics that are simple enough that they are a waste of lecture time.  I can use the leftover time to do more examples.

Problem Set Policies

The best way to learn physics is by doing problems.  In addition to the regular problem sets, Thornton has many problems with full solutions in the Student Solutions companion volume.  Below, I also provide last year's problem sets; solutions are available upon request.
  • Problem sets will be posted below, linked to the syllabus, usually on the Friday or Saturday 1 week before they are due.

  • Due date: Fridays by 5 pm, at my office (311 Downs).  5 pm means 5 pm.  No mercy will be granted.  Remember, we give partial credit, so the last 10 minutes of work will not make much difference.

  • Late policy: Problem sets will be accepted up to 1 week late (5 pm the following Friday) for 50% credit, and after that not at all.  You may turn in part on-time and part late.  Please note on the problem set if it is being split this way.  You do not need to contact me or the TAs to turn in a problem set late at 50% credit, or to turn in part on-time and part late.

  • Extensions:

    • You may take one full-credit one-week extension per term.  No need to contact us, just write it on your problem set.

    • Otherwise, extensions will be granted for good reasons -- physical or mental health issues, family emergency, etc.  You must contact me or one of the TAs before the homework is due and you must provide some sort of proof (e.g., note from resident head, health center, counseling center, or Barbara Green).  A heavy amount of other coursework is not sufficient reason for an extension (though you may use your free extension in such circumstances -- so save it until you really need it!).

  • Solution sets will be posted in the same location when the homework sets are due (usually Friday night or Saturday morning).  If you turn in the problem set late, you may not look at the solutions until you have turned in your problem set.

  • Graded problem sets will be available roughly 10 days after they are due, outside my office.
In spite of my best efforts, sometimes I make mistakes in assigning problems; perhaps not providing enough information, or giving a problem that results in an algebra nightmare.  I will post corrections on this web site, highlighted in boldface at the top of the page and in the syllabus below.  If you are having trouble with a problem, be sure to check this page to see if a correction has been posted, and feel free to contact me if you think a problem has errors in it or seems overly difficult.


The course grade will be one-third homework sets, one-third midterm, and one-third final.

Collaboration is permitted on homework sets, but each student's solution must be the result of his or her own understanding of the material.  No manual xeroxing is allowed.  See below for some comments on working in groups.

Use of mathematical software like Mathematica is allowed, but will not be available for exams.  Prof. Mabuchi makes a very good point on his Ph125 web site: It is absolutely essential that you develop a strong intuition for basic calculations involving linear algebra, differential equations, and the like.  The only way to develop this intuition is by working lots of problems by hand; skipping this phase of your education is a really bad idea.  Be careful how you use such packages.

The midterm and final are not collaborative, though you are welcome to consult your own notes (both in-class and any additional notes you take), Hand and Finch, and my lecture notes (including typo corrections).  Thornton is not allowed because it is not a required text.  You may not use other textbooks, the web,  any other resources, or any software of any kind.

Ph106/196 option:

For Ph106a, we are going to offer an alternate registration option, Ph196a.  Ph196a students will attend the same lecture as Ph106a students, but will be assigned extra reading material covering advanced topics (which will not be covered in lecture), will be assigned more difficult problems for homework sets and exams, and will be graded more rigorously.  This option is being provided to address the wide range of readiness and ability in students taking Ph106a; in past years, we had a very large spread, making it difficult to provide a course that met everyone's needs.

I emphasize that there is no disgrace in taking the 106a option.  You have had only 1 quarter of mechanics by this point in your undergrad education, and expecting everyone to get up to speed on all the subtleties of analytical mechanics in one term is simply unreasonable.  A 106a-level class is typical of most universities.  The class I took as an undergrad was much more like 106a than 196a; I don't think I would have been ready for 196a.

If you just want to register for Ph196a, simply complete an add/drop form as if you were doing a standard class drop and add.  It is possible to register for Ph196a now and drop into 106a later if you are not happy with your performance.  Dean Hall says that  you should sign up for Ph196a initially.  If you later want to drop down to Ph106a, it will be treated as a section change.  You will have until drop day to do this.   You should not attempt to register for both courses, so you will not need to worry about making an overload petition.  Note that we will use your Ph196a problem set and exam grades up to the point at which you switch, so there is a risk in starting in Ph196a.

This policy is being continued into winter term as Ph106b/Ph196b.  Whether it continues to spring term will be decided by Prof. Cross.

Grade Distributions and Anonymously Listed Grades

Histograms of grades for the problem sets to date can be found here (updated 2007/03/29). 

You can check that we have the correct grades recorded for you here (updated 2007/03/29).

Syllabus and Schedule, Lecture Note References, Problem Sets, and Solutions

Boldface: major topic for day
Normal typeface: specific topics to be covered.
Italicized typeface: Ph196 topics; Ph106 students will not be responsible for italicized topics.

LN = Lecture Notes
HF = Hand and Finch
Th  = Thornton (Thornton reading always optional)

Tuesday Class
Thursday Class
Friday Problem Set
Fall, 2006
Sep 26
review of Newtonian mechanics
LN 1.1.1-1.1.2
(Th 2.1-2.5)
Sep 28
review of Newtonian mechanics

LN 1.1.3 and 1.2
(Thornton Ch. 2.6, 5)
Sep 29
Problem Set 1 posted
v. 2 posted 10/1
Oct 3
dynamics of systems of particles
LN 1.3.1
generic results
(Th. 9.1-9.5)
Oct 5
dynamics of systems of particles
virial theorem
LN 1.3.2
(Th 9.6-9.8)
Oct 6
Problem Set 1 due

solutions posted

(v. 2!)
Problem Set 2 posted
Oct 10
of Lagrangian mechanics

constraints and generalized coords
nonholonomic constraints
virtual work
generalized equation of motion
LN 2.1.1-2.1.3
HF 1.1-1.7

Oct 12
of Lagrangian mechanics

conservative forces and Lagrangians
Euler-Lagrange equations
solving problems with E-L eqns
LN 2.1.4-2.1.8
HF 1.8-1.11
Oct 13
Problem Set 2 due
solutions posted
Problem Set 3 posted
v. 2 posted 10/15
Oct 17
special topics in
Lagrangian mechanics

Lagrangians for nonconservative forces
symmetry transformations
Noether's theorem
LN 2.1.9-2.1.10
HF 1.11-1.13, 5.1-5.2
(Th 7.9)
Oct 19
variational dynamics, incl. constraints
calculus of variations
principle of least action and E-L eqns
Lagrange multipliers for
holonomic constraints
LN 2.2.1-2.2.3
HF 2.1-2.7
(Th 6.1-6.5, 7.1-7.7)
Oct 20
Problem Set 3 due
solutions posted
Problem Set 4 posted
v. 2 posted 10/25
Oct 24
constraints in variational dynamics
Examples using Lagrange multipliers
Variational techniques for
nonholonomic constraints
LN 2.2.3-2.2.4
HF 2.6-2.8
(Th 6.6, 7.5)
Hamiltonian dynamics
Legendre transformations
Hamiltonian dynamics -- theory
Hamiltonian dynamics -- examples
  LN 2.3.1
HF 5.3-5.5
(Th 7.10-7.11)
Oct 26
advanced topics in
Hamiltonian dynamics

phase space
Liouville's theorem
Virial theorem
LN 2.3.2, 2.4.1
HF 5.6, 6.1-6.2
(Th 7.12)
Oct 27
Problem Set 4 due
solutions posted
v. 2 posted 11/10
Midterm posted
Oct 30
advanced topics in
Hamiltonian dynamics

canonical transformations
symplectic notation
Poisson brackets
 generating functions
action-angle variables and adiabatic
invariance via generating functions

LN 2.4.2-2.4.4
HF 6.3, 6.5
Nov 2
advanced topics in
Hamiltonian dynamics

Hamilton-Jacobi theory
and examples

LN 2.4.5
HF 6.5
Nov 3
Midterm due
 solutions posted
(v. 2 posted 2006/11/07)
Problem Set 5 posted

Nov 7
action-angle variables
adiabatic invariants
simple harmonic oscillators
simple harmonic oscillator
damped and driven SHO
Green's functions for SHO
LN 3.1.1-3.1.4
HF 3.1-3.7
(Th 3.1-3.5, 3.7-3.9)
Nov 9
simple and coupled
harmonic oscillators
Green's functions for SHO
SHO resonance phenomena
finding normal modes
LN 3.1.5, 3.2.1-3.2.2
HF 3.8-3.9, 9.1-9.3
(Th 3.6, 12.1-12.4)
Nov 10
Problem Set 5 due
solutions posted
solutions updated 12/3
Problem Set 6 posted
v. 2 posted 11/16
Nov 14
No lecture
(observing galaxy clusters,
watching the virial theorem in action)
Nov 16
coupled oscillations
applying initial conditions
mathematical structure
degenerate modes
loaded string
continuous string
LN 3.2.2-3.2.4, 3.3.1-3.3.3
HF 9.4-9.7
(Th 12.5-12.9, 13.1-13.?)
Nov 18
Problem Set 6 due
solutions posted
solutions updated 12/4
Problem Set 7 posted
You have 2 weeks for it. 
Plan accordingly!
Nov 21
central force motion
effective 1-D eqn of motion
qualitative dynamics
Kepler's 2nd law
formal solution
LN 4.1.1
HF 4.1-4.4
Nov 23
Thanksgiving holiday
Nov 24
Nov 28
central force motion
1/r2 central force solutions
scattering and cross sections
LN 4.1.2
HF 4.5-4.7
Nov 30
 Review for final

Dec 1
Problem Set 7 due
solutions posted
v. 3 posted, 1/15
Final exam posted
Final exam due Dec 8
solutions posted
v. 2 posted, 1/3

Winter, 2007
Jan 2
No class
Jan 4
mathematical description of rotations
infinitesimal and finite rotation matrices
Lie algebras and groups
LN 5.1
HF 7.1-7.4
(Th 1.2-1.8)
Jan 5
Problem Set 8 posted
v. 3 posted 1/12

Jan 9
rotating coordinate systems
obtaining an effective eqn of motion
LN 5.2.1-5.2.2
HF 7.5-7.8
(Th 10)
Jan 11
rotating coordinate systems
dynamics of rigid bodies
mathematical description of
rigid body motion
LN 5.2.3, 5.3.1
HF 7.9, 8.1-8.3, 8.7-8.8
(Th 11.1-11.8)
Jan 12
Problem Set 8 due
solutions posted
Problem Set 9 posted
 v. 3 posted 1/18
Jan 16
dynamics of rigid bodies
dynamics of torque-free rigid bodies
LN 5.3.2
HF 8.4-8.6
(Th 11.9-11.10)
Jan 18
dynamics or rigid bodies
dynamics of rigid bodies with torque
LN 5.3.3
HF 8.10
(Th 11.11)
Jan 19
Problem Set 9 due
solutions posted
Problem Set 10 posted
Jan 23
special relativity
basic postulates
transformation laws
LN 6.1.1-6.1.2
HF 12.5-12.8,
read only the first two subsections of 12.6
Jan 25
special relativity
physical implications
Lagrangians and Hamiltonians in SR
LN 6.1.4-6.1.5
HF 12.9-12.12, 12.14
Jan 26
Problem Set 10 due
solutions posted
Problem Set 11 posted
Jan 30
special relativity ctd
mathematical description
LN 6.1.3
classical field theory
continuous string example
generalized calculus of variation
EOM for nonrelativistic fields

Feb 1
classical field theory
Noether's theorem
Hamiltonian mechanics
gauge transformations
Feb 2
Problem Set 11 due
solutions posted
Midterm posted
Midterm due Feb 9
solutions posted


Exam Parameters: All exams will be 4 hours, to be done in one sitting, but with a total of 30 minutes of break time allowed.  Policies on what materials you may use are given above.

Midterm, Fall term: This exam will cover material through Oct 19 and in addition the material on Oct 24 on using Lagrange multipliers to implement constraints in variational dynamics.  Topics that Ph196 students will be responsible for but will not be required for Ph106 students:
  • virtual work and generalized forces
  • derivation of Euler-Lagrange equations via virtual work
  • nonholonomic constraints
  • Lagrangians for nonconservative forces
  • Dealing with nonholonomic constraints using Lagrange multiplier methods
Exam instructions (1 page): Download these first!
Exam (including instructions, 3 pages): Don't download this until you are ready to take the exam.
Solutions (typo on Problem 4 soln corrected, 2006/11/07)

Comments: this exam was too easy by about 10 points.  On problem 4, one could miss subtle parts of the problem with losing a lot of points.

Final, Fall term: This exam will cover all material from the term.  In addition to the above, the 196-only topics are:
  • Legendre transformations
  • canonical transformations and generating functions
  • symplectic notation, Poisson brackets, action-angle variables, and Hamilton-Jacobi theory
  • mathematical structure of coupled oscillations
Exam instructions (1 page): Download these first!
Exam (including instructions, 4 pages): Don't download this until you are ready to take the exam.
Solutions (Version 2 posted, 2007/01/03).

Comments: This median score was 62 with rms 26.  I was hoping for a median of about 70.  This was primarily the fault of problem 4, for which the median score was 5/20 points; if it had been 13/20 or 15/20, the median would have come out in the right place.  Specific points:
  • Problem 1: the median score was 15/20 points.  That's probably about right -- looks like most people got most of the problem right, missing a few points here and there due to mistakes.

  • Problem 2: The "quickie" problems did not really serve their purpose of being very easy: the median score was 15/20 points, no better than problem 1.  I'm sure many of you will be surprised at how straightforward these are when you see the solutions.

  • Problem 3: This turned out to be the easy one.  So you know coupled oscillations well -- I'm glad to see that.

  • Problem 4: It looks like you either got it or you didn't.  I should have given a hint to make it clear what needed to be done -- find the particles with impact parameter that bring them closer to the planet than its radius R -- which would have pushed the median up a lot.

  • Problem 5: (a) and (c) seemed to cause the most trouble, while (b) was more doable.  This is unfortunate but not surprising, as (a) and (c) required almost no calculation, just physics, while in (b) one could just rely on the procedure for solving an inhomogeneous differential equation and get almost all the points.
Midterm, Winter term: This exam will cover rotating coordinate systems, dynamics of rigid bodies, and special relativity.  Topics that will not be covered are
  • Lie groups
  • Mathematics of special relativity
  • Classical field theory
Exam instructions (1 page): Download these first!
Exam (including instructions, 3 pages): Don't download this until you are ready to take the exam.
Solutions (Version 2 posted, 2007/02/16 to reflect other possible reading of Problem 3).

Comments: This median score was 52, mean 48, rms 26.  A bit of a bloodbath.  Again, I was hoping for a median of about 70.  This was primarily the fault of problem 3, for which the median score was 4/20 points.
  • Problem 1: median score = 14, reasonable.

  • Problem 2: median score = 15, reasonable.  I'm glad that you understand rotating systems.

  • Problem 3: median score = 4.  This was supposed to be easy -- you'll probably be kicking yourself when you see the solution.  I'd appreciate hearing why it was difficult.  I gather that some of the low grades may have been because one can read the problem in such a way as to infer the opposite velocity as intended.  If so, you should get full credit if you did the problem correctly under that interpretation.  A solution for that reading of the problem has been added.  I also have the impression is that there are some somewhat correct solutions that were not given enough credit.  If you feel that your solution is partially or wholly correct and did not receive enough credit, contact Andrey.  If you can't arrive at a satisfactory solution with Andrey, contact me.

  • Problem 4: median score = 10.  This was supposed to be difficult conceptually, looks like it was.

  • Problem 5: median score = 10.  This was largely a matter of Taylor expanding the effective potential.  Not sure what else to say.

Update (2007/03/02): With many of you returning your midterms for correction of grading on Problem 3, the statistics have improved: median 59, mean 47, rms 24.  The median score on Problem 3 is now 12 rather than 4.  This is more reasonable.  We apologize for not realizing earlier how Problem 3 could be read in a different way. 

Practice Problems

Use these problems to check your understanding and develop your physical intuition.  Solution sets are available by email request.

2004-2005 Problem Sets:
2005-2006 Problem Sets:
2004-2005 Exams
2005-2006 Exams
  • 2005 Fall Midterm: Material through variational dynamics
  • 2005 Fall Final: All material through waves and continuous systems
  • 2006 Winter Midterm: Special relativity, central forces, rotating systems, rigid body motion, making use of some material from Fall 2004.

Comment on Working in Groups:

It is in general a good thing to work with other students while reading and doing problem sets.  You get to hear different perspectives on the material and frequently your peers can help you get past obstacles to understanding. 

However, you must use group work carefully.  If you rely on your colleagues too much, or take a very long time to do the homework sets, you will do poorly in the fixed-time, independent exam environment.  Empirically, we observe that students with good exam scores tend to also have done well on homework, but that good homework scores do not predict good exam scores.  Exam scores correlate from exam to exam, even on largely independent material.  For example, scores from 2004-2005 Ph106ab:

Notice, in particular, the midterm-final correlation for Ph106b, which is remarkable because the exams covered totally disjoint material (mechanics vs. E&M) and were written by two different instructors. 

To avoid suffering from this problem, I have two suggestions:
This is not just an arbitrary classroom exercise.  In research, one is always under schedule pressure -- because one only has a fixed number of nights at an observatory, because there are funding deadlines, because there are competing groups doing similar work.  It is critical to learn how to cut through irrelevant or unimportant information and get to results in a timely fashion.

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