Blog Update

I have not posted for a couple of days because I am hard at work on transcribing Lecture 4 of Classical Mechanics. This is about symmetry and conservation laws.

I am also working on a book review for a book I received for review from CRC Press, Advanced University Physics.

Lots on my plate this week.

Book project update

I have just completed a 15 page chapter on calculus that covers limits, derivatives, integrals, differential equations, infinite series, and partial derivatives. This chapter is for the proposed classical mechanics book I am working on with Leonard Susskind.

I took a very minimalist approach to the subject. If you want to see how I crammed so much into so little, wait for the book... :-)

Actually, some of this material will likely find its way into my column (and I took some of it from my existing columns).

The Grand Design - The Straight Story

I have just completed reading, "The Grand Design," by Stephen Hawking and Leonard Mlodinow. It was a good book. I do not agree with everything they write, but then it is for a lay audience and not for specialists. This book is available for sale from the bookstore below.

Let me state one thing right up front and make it completely clear: Nowhere in the book do the authors state that God does not exist! Period! As I suspected (see a previous post) all the authors state is that the current physical theories are such that no appeal to God is necessary to explain the existence of the universe. In other words, the physical laws are rich enough to explain how the universe came into being out of nothing.

The one objection that I have to the book is the reliance on M-theory, this is a theory that I do not understand; so it is difficult for me to critique it. I know just enough to get myself into trouble, so I will refrain.

I enjoyed the book a lot! if you are interested in physics, this is worth getting.

Ignore all the hype, it is nothing but lies! No one who has read the book can truthfully state that the authors said that God does not exist. It is a very nioe history of the development of theories of the universe of all types.

Secret Weapons of The Theoretical Physicist

Tonight I am thinking about a set of the most powerful techniques of the theoretist, classically called back-of-the-envelope calculations. These consitute two broad categories of methods.

The first is the method of estimation, where you use some basic reasoning abilities to determine a rough estimate of the quantities you are considering. For example, you want to estimate the number of hairs on someone's head. You think about it and decide that if you knew the area of the part of the head covered by hair, then knew the average area of a hair, you could figure it out. Then you make the necessary assumptions about the quantities (either by looking up the information, or by figuring it out for yourself).

The second is the ability to check the correctness of equations and to derive new equations by studying the relevant units of the quantities. All terms of any equation must have the same units. When you chaeck the terms, any that do not have the correct units are just wrong. By including powers of each quantity in each term and making sure all of the units come out the same, you invent a system of algebraic equations that can be solved. This gives you the necessary powers of each variable and shows you the structure of each term within a factor of a constant of proportionality. In this way you can learn what each term must look like. This allows you to invent new equations if you know the units involved. This is called dimensional analysis.

I will be writing about these ideas in my column in the coming weeks. I will also be discussing this in the Classical Mechanics book I am writing with Leonard Susskind.

An Apology and a Reminder

I just deleted a comment on a previous post, I will not tell who sent it, or exactly what it was about. It had the broad topic of mysticism and religion. While I am happy to discuss such things, this is not the forum for such discussions.

Please do not send comments about such topics to me. I will not publish them.

If you want to discuss physics, that is fine. If you think you have proved Newton, Galileo, Einstein, Bohr, Planck, Feynman, Hawking, etc. wrong, tell the New York Times, not me! I sincerely doubt that your theory will meet my standards...

If you have interesting ideas about physics, that is fine. I have an open mind, but it is not so open that I am liable to fall in... I also am completely skeptical of wild claims (ask anyone that has been a speaker at a physics colloqium or seminar that I have attended...).

This blog is about theoretical physics education and research. Let's limit it to that.

George

Downplaying the Hawking-Mlodnow "Grand Design" Controversy

People, mostly reporters, have been driving me crazy again. Specifically, they are making statements about the new book, "Grand Design," by Stephen Hawking and Leonard Mlodnow without quoting the book directly. Now, let me say up front that I have not yet read the book. I have heard the statements that Hawking has made in the past along the lines of, "God is not necessary to the creation of the universe." At no point, including in this book, has either author made that statement that God does not exist, nor that God did not create the universe.

Despite this, reports are that they have said these things in their book. Having seen statements made by the authors and listening to interviews with Mlodnow, I cannot believe that people are ignoring the authors!

Come on people! Stop making an issue out of nothing!

Pythagorean Triples

Pythagorean triples are groupings of three integers that each form the sides of a right triangle. I have been playing with their distributions, and have made some plots in Mathematica.  I think these are pretty cool.

The Mind of a Theorist - The Relaunch of the Column

You can find the rewritten and reformated column at www.madscitech.org
I will be placing a new column up every week.

New Version of Mathematica Announced

I was just informed by Wolfram Research that a new version of Mathematica will be coming out soon. As I understand it, this version is going to be a big step forward. This version will have built-in GPU support for CUDA, built-in control systems and wavelets, advanced data visualization tools, and a huge raft of other things I have heard about, but have not seen officially so will keep my mouth (fingers?) shut about.

This is exciting news since there is a personal version of Mathematica for only about $300, with a $500 upgrade to the most current professional version. There is no difference between the personal and profession versions.

Book Review: The Classical Theory of Fields

Time for another book review. You can purchase this book at my book store below.

This time I am going for volume 2 of the Course of Theoretical Physics by Landau and Lifshitz. This volume is titled, "The Classical Theory of Fields," and covers, in one volume, special and general relativity, electrodynamics in a vacuum, and optics in a vacuum. The authors assume you are familiar with vector analysis and the electromagnetic phenomena equivalent to that covered in a general physics course (charges, electric and magnetic fields, and induction).

The Wonders of Physics

Earlier this year, I participated in a public demonstration lecture called the Wonders of Physics. In this year's show I demonstrated the physics of board-breaking.

I had a lot of fun, and you can find a video from a friend of mine, showing my participation in one of the earlier shows: http://www.youtube.com/watch?v=4eQWRLZG1rU

The official web site for the Wonders of Physics, where you can see all of the shows over the years, is here: http://sprott.physics.wisc.edu/wop.htm

I had a lot of fun, and it is a nice to give such presentations to the public, who hopefully will become interested in physics.

Book Review: Mechanics, by Landau and Lifshitz

It is my intent to review every book in my store, below, so I will start with the first book in the store. You can purchase this book at my book store below.

This is the first volume of the classic Course of Theoretical Physics due to Lev Landau and the Landau Institiute of Theoretical Physics in Russia. These ten volumes were the subject of what Landau belived to be the required preparation in physics for any theoretical physicist.

This volume begins with a touching introduction to Lev Landau and his philosophy. It also lays out the prerequisites for this volume. To quote: "...ability to solve any indefinite integral that can be expressed in terms of elementary functions and to solve any ordinary differential equation of the standard type, knowledge of vector analysis and tensor algebra as well as the principles of the theory of functions of a complex variable (theory of residues, Laplace method). If you need this ackground I recommend Hassani, Mathematical Methods for Students of Physics and Related Fields, Second Edition.

The first chapter describes Lagrange's equations of motion from the starting point of generalized coordinates. This lays the ground work for a very nice presentation of the least action principle. Then there is a nice section of the Galilean transformations; it is important to realize that there is no discussion of special relativity in the book. The book then describes the Lagrangian of a free particle in Cartesian, spherical, and cylindrical coordinates. The final section of the first chapter not only discusses the Lagrangian of a system of particles, but nicely links Lagrange's equations to Newtonian theory.

The second chapter presents one of the most important aspects of theoretical physics, the conservation laws. This begins with a unified treatment of integrals of the motion, the conservation of energy, how conservation of energy implies the homogeneity of time, and conservative systems. This leads into the conservation of momentum, how the conservation of momentum implies the homogeneity of space, generalized momenta, and generalized forces. There is a brief discussion of systems of particles, the center of mass, and internal energy. This leads to a section on the conservation of angular momentum, how this implies the isotropy of space, and the idea of the central field. The final section deals with a couple of different topics such as the behavior of the equations of motion under transformations and the virial theorem.

The third chapter is a very practical one about applying the equations of motion to specific situations. This begins with one dimensional motion and a discussion of energy diagrams and oscillations. This leads into a discussion of how to interpret energy diagrams and derive the potential from a specific period of oscillation. The next topic covered is the beginning of the two-body problem in the form of the reduced mass of a system. The two-body problem is then reduced to a one-body problem in a central field in a very clear presentation. This leads to a nice discussion of the Kepler problem for bound and unbounded orbits.

Chapter four is an important discussion of the classical theory of particle scattering. This begins by the idea that particles can break up, this also introduces the lab and center-of-mass frames of reference. Then the authors describe elastic collisions in quite intuitive way. Then the notions of impact parameters and scattering-cross sections are covered. As an example of scattering due to fields, there is a section on Rutherford scattering. The chapter ends with small-angle scattering.

Chapter Five is on the theory of oscillations. Naturally, this begins with free oscillations in one dimension. The book then turns its attention to forced oscillations and resonance. Then the authors treat the idea of oscillations in more than one degree of freedom, including eigenfrequences and normal modes. This is then applied to the classical theory of molecules. Then damped oscillators are covered along with a discussion of dissipative functions in Lagrangian dynamics. The next section introduces dissipation and damped and driven oscillations. At this point the book leaves the realm of simple oscillating systems and goes into parametric oscillations and the Mathieu equation. There is a discussion of nonlinear oscillators, laying the ground-work for the study of chaos, this also includes a section of the resonance of nonlinear oscillators—introducing the Duffing oscillator without calling it that. Finally, there is an interesting section on the motion of a particle in a rapidly oscillating field.

Chapter six is a thorough treatment of rigid body dynamics. This begins with a discussion of the kinematics of rigid bodies. Then the authors introduce the inertia tensor. Indeed, this is one of the few sections in the book with more than a few problems (there are nine here). The section on angular momentum uses the axes of inertia to define angular momentum. Then the equations of motion for a rigid body are derived. This leads to the notion of Euler angles, which in turn leads to the Euler equations. These notions are then applied to the motion of a top. Then they shift gears a little and discuss rigid bodies in contact. Finally, and I think this might have done better at the beginning of the chapter, is a section on motion in noninertial frames, specifically rotating frames. This chapter seems very conventional, though it must be remembered that the first version of it came out in 1960, so a lot of other books are based on material found in here.

The final chapter is an overview of Hamiltonian theory. This begins, reasonably enough, with Hamilton's canonical equations of motion and relates the Hamiltonian and the Lagrangian in the normal way. The Routhian has its own section. Then they address the extremely important topic of the Poisson brackets. The next section establishes the idea of a functional without using those words. Then the authors discuss the principle of Maupertuis. Then they discuss canonical transformations and link them to Poisson brackets, and discuss conjugate variables, laying some of the groundwork for quantum mechanics. The authors next introduct the idea of phase space through a discussion of Liouville's theorem. The next section discusses the Hamilton-Jacobi equation and the idea of general and complete integrals. This is followed by a treatment of the method of separation of variables. Then the authors have an excellent and clear discussion of adiabatic invariants, a subject with applications in plasma physics. Then they discuss action-angle variables. Then the authors turn to the validity of the adiabatic invariant. The chapter, and the book, ends with a section on conditionally periodic motion.

The book, with all problems included, is just 167 pages long and comprises a total of 52 sections. This is something you could easily cover in two months of dedicated study. I think the book could use an update, but it's a classic as it is and its choice of topics are pretty good. I like it a lot, even if it doesn't really cover chaos in any meaningful way.

Conservation of Energy and The Nature of Forces

Today I have been toying with a bunch of things:

1) Conservation of energy, I have been writing this section for the book project. Nice approach, if I do say so myself. Did you know that we really don't know what energy is other than a calculated number that remains the same in a closed system? We have no better understanding of energy than that. Despite being able to calculate it, predict it, and use it in other kinds of calculations; we have no idea what energy IS!

2) In Newtonian mechanics the entire game is chasing the forces. We have F=ma, and people think this is the definition of force. No! This states that the numerical quantity of force is equal to the product of mass and acceleration. If you do not know what F is prior to doing a calculation you will not be likely to figure it out.

3) I have been messing around, playing with trigonometric and hyperbolic functions.

Been having a lot of fun with these things.

Brief on Calculus and Appreciation to Leonard Susskind for his Lectures

Hello everyone!

I have been working on the book project with Leonard Susskind. I just completed an 11-page overview of calculus with detailed material on limits, derivatives (including a short table of derivatives), integrals (with another short table of integrals), differential equations (covering mostly separation of variables), and Taylor series.

I am thinking of adding sections on partial derivatives, maxima and minima, and maybe curvature.

Of course, Leonard has not committed to anything yet; I certainly have not produced enough material to do more than whet his appetite for more. It is possible that something could happen to prevent me from finishing, and he probably does not want to stick his neck, or his reputation, out for someone who is not too well known.

I am having great fun writing this material. It is extremely challenging to develop short and clear explanations for these topics. My olf Mind of a Theorist column is good for this, the philosophy that Iadopted for the column is the same the Leonard has in the lectures.

To quote from the introduction, "What is this course about? Who am I teaching to? While undergraduate or graduate students will be able to read this, it is mostly designed for people who are interested in getting into the meat of physics right away. This is not a standard physics course. This is the real deal, theoretical physics at full scale! We use equations, and sometimes hard equations, but we try to use the simplest equations that will do the job. Basically, we try to keep it minimal. The goal here is to get to the basic ideas fast. So we will be telling you what you really, really, need to know to get to the next level. Sometimes the basics can be hard; we will do them anyway, but will only spend the minimum time required to get them right. And here we mean getting them really right, not by metaphors or analogies, but equations when necessary."

I am dazzled by the structure of the video lectures. In lecture one he introduces the ideas of dynamical systems, phase space, and conservation laws with no mathematics; just some diagrams and very clear explanations. Wonderful!

Wonderful Opportunity

I have begun converting Leonard Susskind's video lecture series on Modern Physics to book form. I have just completed Lecture 1 of the video course Modern Physics: Classical Mechanics and am now working on a chapter on calculus.

So far, so good. These are still early days, but I plan to convert them all in time.

Whether it becomes anything is, at this point, up in the air; we will see how it goes.

I am honored to be working with someone so passionate about providing a good resource on theoretical physics to enthusiasts. I am honored that he didn't just slam the door in my face! I am having a great time in converting these lectures, not only because I get to work with one of the giants of theoretical physics; but also because I am forced to think deeply about the meaning of the lectures and their context. It is a wonderful opportunity.

Thanks, Leonard!

Great Find!

Dover Publications has produced a version of Richard Feynman and Albert Hibbs' excellent textbook, "Quantum Mechanics and Path Integrals."  The text has been cleaned of typographic and grammatical errors, and the notation has been standardized, while maintaining the rough edge to the original. This is probably the best text for the path integral approach to quantum mechanics. Enjoy!