First we’ll finish the remains from the previous class:
- the last part of “Why Mathematics Is So Effective in Physics”, and
- the second half of the lecture on the principle of least action.
Then we’re going to work out some problems in class using the principle of least action, also called Hamilton’s Principle. This will involve correctly setting up the Lagrangian for each problem and working out the equations of motion using the Euler-Lagrange equations. We will work these problems out in detail, omitting little algebra. I’ll try to have hardcopies of this material available at class time.
Then we’ll take up Lecture 7, Symmetries and Conservation Laws. In this lecture Susskind defines symmetries and shows how they lead to the conservation of various quantities. Near the end of the lecture we’ll work on a few problems, just as we did above.
Assignment: Read Lectures 6 and 7 in the book. If time allows, view Lecture 3.
I’ve posted the worked examples for lectures 5 and 6 from the book to our document repository. (Lecture 4 had no examples.) The file is named “Exercises problemsD.pdf” and is available through this link.
This article from the London Review of Books offers a review of Newton and the Origin of Civilization by Buchwald and Feingold. The review offers a short summary of Newton’s career with particular attention paid to his life-long interest, or obsession, with biblical studies. The article is reasonably short and quite interesting.
We’ll start, of course, by finishing anything left over from the third class.
We’ll try to cover
- Interlude 3, Partial Differentiation, which includes extrema,
- Lecture 4, Systems of More Than One Particle, which includes momentum and phase space, and
- Lecture 5, Energy.
If time allows we’ll begin Lecture 6 from the book, where formal classical mechanics really starts. If we do begin this, I’ll keep it self-contained so you needn’t read ahead (unless you’re so inclined). You should consider reading Lecture 6 in the book and viewing Susskind’s Lecture 3 before our fifth class, on October 21, 2013.
I’ve posted the answers to the exercises in Lecture 3 (dynamics) and Mathematical Interlude 3 (partial differentiation) in our document repository. The file is named “Exercises problemsC.pdf”. The pdf file is directly available here.
Last class we almost finished differential calculus. In the next class (October 7), we will aim to
- finish differential calculus, covering the short topic of Composition Rules;
- discuss the kinematics of motion, that is, the quantities we use to specify the path an object takes in regular, physical space;
- discuss integral calculus, which is how we compute the area under a curve; and hopefully
- begin particle dynamics (the real beginning of physics), which comprises the rules that tell us how forces control motion.
Completing dynamics will take us through the end of Lecture 3 in the book (page 73 in the hardcopy version), so you should take a shot at reading that far. We will still be in Susskind’s 2nd lecture. If your time allows, viewing or reviewing that lecture should be helpful.
The next worked exercises will cover particle dynamics and partial differentiation. I’ll try to have those posted by the next class time (October 7) or shortly after.
I’ve posted the answers to the exercises in Lecture 2 (motion plus differential calculus) and Mathematical Interlude 2 (integral calculus) in our document repository. The file is named “Exercises problemsB.pdf”. I’ve done at least one problem from each set. The pdf file is directly available here.
If you go to the bottom of the Resources page, you’ll find a link to a set of solutions posted by George Hrabovsky, the second author of our text. (Thanks to Gladys for reminding me of this.) I have trouble viewing his pdfs for unknown reasons and I’ve only hacked my way through a few of them. You may have better luck than I did.
If you have questions or comments about either my solutions or Hrabovsky’s, be sure to raise them in class.
I’ve decided I’m going to try to keep up with the exercises (time permitting). I’ve worked through the exercises for Lecture 1 and Interlude 1 (which is what we’ve covered so far except for differential calculus and motion). They’re available on our document repository as a pdf file that starts with the word “Exercises”. (For your convenience, this link should take you directly to the pdf file. I hope.)
I’ll try to get through Lecture 2 (differential calculus and motion) before the week is out.
If you find some possible errors let me know. If you do it by commenting on this post, it will let all of us share in the discussion. Don’t be shy about raising topics you find confusing; others probably feel the same way.
In the early 60’s, the late Richard Feynman, a Nobel Prize-winning physicist at Caltech, gave a two-year course of lectures on physics for Caltech freshmen and sophomores. These lectures were edited into a three-volume set that was used thereafter as the text for Caltech’s lower-division (universally required!) physics courses. Feynman’s course was taken simultaneously with two years of calculus.
I took the course the first year the textbooks were in print. They were different from any physics text I have seen before or since: insightful, entertaining, and hard on undergraduates. My later impression was that the books were personally very popular with physicists but were not widely used as texts because of their difficulty.
The first volume of the series, which covers mechanics, is now freely available on the Internet at http://www.feynmanlectures.caltech.edu/. The first three chapters include his overview of physics as of 1963, which is not radically different a half-century later (so far as I know). Chapters Four through Fourteen cover mechanics and are interesting to sample; note particularly Chapter Eleven on vectors. Chapter Sixteen, Section One, has his famous complaints about “cocktail-party philosophers.”
The title of this post is taken from a New York Times opinion piece, here, from September 15, 2013. The piece was written by a mathematician who encourages us to view mathematics as more than a utilitarian tool for application in other fields (such as physics).