# Physics 105 - Analytic Mechanics

[shortcut to the homework assignments]

## Basic Info

Fall 2004, Tue and Thu, 8:00 - 9:30 a.m., 60 Evans Hall

Instructor: Petr Hořava (email: horava@socrates.berkeley.edu)
Office: 441 Birge (office hours: Thu 10:00 - 12 noon, Thu 1:30-2:30 p.m.; usually, you can also find me there on Tue, 10:00 -12:00 noon); 50A-5107 LBNL (usually on Mon, Wed, Fri)

Discussion sections: Tue 1:00 - 2:00 p.m. 4 Evans, Wed 4:00 - 5:00 p.m. 6 Evans
GSI office location and office hours: Mon 2:30 - 3:30 p.m. in 397 Le Conte,
Wed 5:00 - 7:00 p.m. in 395 Le Conte.

## New/Important Announcements

(12/22): The final grades have been posted on bearfacts.

(12/21): The official solution of the final exam has been posted.

(12/21): I am available to discuss the results of the final in my office (441 Birge) today, Tuesday Dec 21, 10am-12:30pm, 1:30pm-2-10pm, 2:30pm-4pm. If you cannot come but you wish to discuss your results, please send me an email.

(12/21): Results of the final: Overall, the results were good, everyone with the exception of two people did better on the final than on the midterm. Here are some facts:

Total number of possible points: 50 + 5 bonus
Number of people who attempted the bonus problem: 0
Highest score: 43.5
Lowest score: 6
Median score: 29 (pretty good!)
Total final grade: For almost everyone, the grading was based on the formula 50% homework, 50% final. (Your total homework score was the average of your ten best homework scores.) Here is the breakdown of various grades versus total percentages:
82 -100% : A+
75-81.5% : A
68-74.5% : A-
61-67.5% : B+
54-60.5% : B
47-53.5% : B-
40-46.5% : C+
33-39.5% : C
26-32.5% : C-
25% or less: D
The highest score was 88% (two students tied, actually), the lowest score 20%, and the median score 71% (which corresponds to the median grade of A-). For those students taking the course on the pass/fail basis, if your score corresponds to a passing grade, you pass; i.e., you need at least 26%.

(12/20 midnight): The final exam has been graded, and final grades have been assigned (but have not been posted at bearfacts yet).
More details about the overall results of the final, and about the final grading, will appear here within the next 12 hours (see above).

(12/15): The final exam will be a closed-book, no-electronic-devices, no-notes exam.

(12/5): The list of most important concepts in Chapters 1-9 of [Hand-Finch] has been posted. This list is intended to serve as a useful tool in the preparation for the final exam.

(11/29): The final exam will cover Chapters 1-9 of [Hand-Finch], in a format similar to the midterm exam. The final exam is scheduled to take place on Friday, December 17, 12:30-3:30 p.m., in 390 Hearst Mining.

(11/9): There will be no class on Thursday, November 11 (Veterans' Day).

(11/5, 10:45 p.m.): The official solution to the midterm exam has been posted.

(11/4): Those people who still haven't picked up their graded midterms should do so personally in my office, 441 Birge, either on Tuesday 10-12, 1-2, 3:30-4, or on Thursday, 10-12, 1:30-3 (if these time windows are all impossible for you, please arrange an appointment by email).

(11/4): The overall performace on the midterm exam was relatively very low. It seems that for many students, the problem was in the time pressure they felt, given the number of problems on the midterm. In response to discussions with most students, the breakdown of percentages towards the final grade has been changed, in the following way. Option A stays the same as originally anounced (50% homeworks, 20% midterm, 30% final exam), and Option B will be:

50% homeworks,
50% final exam.
For each student, the final grade will be based on the one Option that gives him/her the higher score. Thus, those students who demonstrate their good understanding of the material at the end of the semester by doing reasonably well on the final, will not be penalized for not performing so well on the midterm.

(10/26): In response to popular requests, and as a result of the vote taken today in class, the due date of homework assignment 7 has been moved to Fri, Nov. 5, 5 p.m. The remaining schedule of homeworks and their due dates for the rest of the semester have been posted on the homework webpage.

(10/22): Midterm grades have been posted on bearfacts. Only those students who are potentially failing this class (i.e., grades D or F) have received grades. If you have received a grade and are interested in finishing this course successfully, you should discuss your situation individually with me (and/or Nadir).

Homeworks will be assigned every Thursday (in the form of a pdf file posted on this web site by 12:30 p.m. each Thursday), and will be due in eight days, on Friday by 5pm sharp, in the corresponding box located in the 2nd floor breezeway between Le Conte and Birge Halls, unless stated otherwise. The schedule of the remaining homework assingments and their due dates can be found on the homework assignments webpage. Solutions of homework problems will be posted on the homework webpage very shortly after the due date of the homework, and absolutely no credit will be given to homeworks that are turned in late. Each individual problem of each homework assignment will be graded on the scale of 0-10 points. If you mess up on one homework assignment, don't despair: In the final evaluation of your total score from all homeworks, we will drop one homework assignment with the lowest score.

Office-hours policy: While I am happy to address any general questions you may have during office hours, as a matter of principle I will not discuss the specifics of any of the homework problems before they are due.

Mid-term exam: Will be administered in class on Thursday, October 28. The exam will start at 8:10 a.m., sharp. It will be a closed-book (and no laptop, etc) exam, based on Chapters 1-5 of the textbook. The problems will be of two kinds: Some will resemble the problems that have appeared on Homeworks 1-6, while some will be more conceptual and will ask about issues more directly discussed in the text of (Chapters 1-5 of) [HW]. (In fact, in your preparation for the midterm you should primarily focus on your thorough understanding of the material of Chapters 1-4; any questions related to the material of Ch. 5 will be very elementary, and will not go beyond deriving the Hamiltonian (and the Hamilton equations of motion) from a given Lagrangian, and understanding some simple aspects of the Noether theorem.)

50% of the final grade will be based on homework assignments,
20% will be based on the single mid-term exam, and
30% will be based on the final exam (based primarily on Chapters 1-9 of the book).
50% homework,
50% final exam.
For each student, the option that gives her/him the higher score will be used.

We will decide on the offical breakdown of percentage versus grade later this semester, and post the official rules in due time.

## Textbook Information

[An approximate syllabus, to be refined, can be found here.]

The primary required text that I will be using for the course is

L.N. Hand and J.D. Finch, "Analytical Mechanics" (Cambridge, 1998)
ISBN: 0 521 57572 9 paperback (0 521 57327 0 hardcover)

This text is based on a similarly advanced undergraduate course taught in recent years at Cornell. This book indeed reflects very closely the logic of Analytic Mechanics as I see it. In my opinion, we are very lucky to have a really good book which can be followed pretty much linearly (at the approximate speed of one chapter per week).
Other books of interest include the following:

J.B. Marion and S.T. Thornton, "Classical Dynamics of Particles and Systems" (4th Edition, Harcourt Brace, 1995)

This book is sometimes used by others who teach upper-division Mechanics, but in my opinion, it is an unfortunate book to use at this level. It spends way too much time repeating Newtonian concepts and takes too long before it moves to the more central areas of Lagrangian and Hamiltonian formalisms. It could, however, be useful to those students who feel that they need to refresh their memory and understanding of elementary Newtonian mechanics.
H. Goldstein, "Classical Mechanics" (2nd Edition, Addison-Wesley, 1980)
Goldstein is a classic text that is always a good book to have in your library. If you compare the Table of Contents of Golstein you will find out that its logic is actually quite close to that of Hand-Finch. On the other hand, Goldstein was written in the 50's and it shows...
There are several other textbooks that deserve an honorary mention, and you may want to take a look at them depending on your interests. Let me mention a few:

L.D. Landau and E.M. Lifshitz, "Mechanics" (3rd Edition, Butterworth-Heinemann, 1981)

This is an excellent, very dense and very no-nonsense summary of basic principles of analytic (="theoretical") mechanics. It is also Volume 1 of the famous Landau-Lifshitz encyclopedic series on theoretical physics, demonstrating the unique position of analytic mechanics as the most fundamental building block of all of modern theoretical physics.
J.V. Jose and E.J. Saletan, "Classical Dynamics. A Contemporary Approach" (Cambridge, 1998)
As I will claim repeatedly during this course, the natural language of analytic mechanics is that of modern differential geometry. However, many of the classic textbooks (Goldstein, Landau-Lifshitz, etc) don't really do a good job with the more geometric aspects of mechanics. Jose-Salentan is meant to remedy that problem, and you can think of it as a sort of modern version of Goldstein, with emphasis on geometrical methods and modern geometrical thinking.
R. Talman, "Geometric Mechanics" (John Wiley and Sons, 2000)
Another book stressing the modern geometric aspects of mechanics.
If you are very mathematically oriented, there are also two classic books on the formal geometric aspects of mechanics,

R. Abraham and J.E. Marsden, "Foundations of Mechanics" (Benjamin/Cummins, 1978), and
R. Hermann, "Differential Geometry and the Calculus of Variations" (Academic Pres, 1968).
They are both excellent (although slightly outdated) but I wouldn't recomment them to people who are not serious about abstract differential geometry.

In addition, if you read French or Russian, there is an excellent little book about mathematical aspects of Analytic Mechanics,
C. Godbillon, "Geometrie Differentielle et Mecanique Analytique" (Hermann, Paris, 1969).
(This book was also translated in the 1970's into Russian and published by one of the main Soviet publishing houses in Moscow.)

A really entertaining advanced presentation of mathematical aspects of mechanics can also be found in the modern classic written by Arnold,
V.I. Arnold, "Mathematical Methods of Classical Mechanics" (Springer, 1989).
Of course, this is an English translation of the Russian original, as published originally by Nauka in Moscow in 1974.

horava@socrates.berkeley.edu