Archive for the ‘Classical Physics’ Category

Lectures on Classical and Quantum Physics

December 13, 2009

The Indian Institute of Technology Madras has some online collections of lectures. Among them, a Quantum Physics course imparted by Prof. V. Balakrishnan. You can see the first lecture below, devoted to an introduction to the conceptual underpinnings of quantum mechanics and, more specifically, Heisenberg’s principle. Prof. Balakrishnan does a remarkable job at explaining these horny issues, avoiding common pitfalls; and, as the students’ questions make him wax philosophical, he shows a very refreshing honesty in not trying to sell the theory as the end of the path, avoiding the “shut up and calculate” stance that was so common during the second half of the past century (and which i still suffered during my undergraduate and graduate years). The exposition doesn’t use any advanced mathematics and keeps at a conceptual level, and i think it should be understandable by anyone with a very modest background (perhaps some acquaintance with classical mechanics will help at some point, but it’s not a requirement to enjoy the lecture).

There’re are thirty more lectures in the course, all of them available for your learning pleasure. Although i haven’t had the time to watch them all, if Prof. Balakrishnan keeps their quality as high as in the first installment, i’m pretty sure they’ll make up for many hours of fun.

And, if you feel like learning classical physics, i’ve got good news for you too: there’s also a series by Balakrishnan on Classical Physics! Here’s the first lecture:

and here you can find 37 more!

The dimensionality of the world

July 17, 2006

Although Bee has recently written an amazing and thorough article over at Backreaction with virtually everything one needs to know about extra dimensions in physics, let me add a sort of footnote in the form of some naive musings, a couple links and a Hertzian digression in this somewhat iffy post.

Multiple dimensions and the problem of time
As a student, i was in love with Kaluza-Klein theory and its extremely elegant explanation of electromagnetism as the purely geometrical effect of a fourth spatial dimension. The really magic thing is that the electromagnetic energy momentum tensor (in four dimensions) arises as a consequence of an empty five-dimensional space where particles follow geodesics; in other words, photons are purely geometry, just as gravitational forces. The problem, of course, was to explain why we don’t measure that fifth dimension. Kaluza just prescribed that no physical quantity depended on it, while Klein tried a somewhat more satisfactory solution by compactifying it to an unobservable size, and making it periodic, just as the second dimension of a long hose, which becomes one-dimensional when seen from a distance. Unfortunately, this beautiful picture seemed to lead to insurmountable difficulties with chirality or the mass of the electron, unless one goes the string way and adds more compact dimensions to our universe. I thought Kaluza-Klein theories were all but abandoned in their original 5-dimensional form these days, but following some links in the recent review article by Orfeu Bertolami, The Adventures of Spacetime, proved me utterly wrong. There’s been quite a lot of activity in the area during the last decade, leading even to a Space-Time-Matter consortium, a sort of physicists’ club promoting 5-dimensional gravity theories without compactification. The consortium is coordinated by P.S Wessan, and has quite a few members and interesting publications: see for instance this comprehensive review of KK theories of gravity for an introduction to Wessan and friend’s ideas. What i find compelling about their approach (and what, at the same time, of course reveals my prejudices) is that they tackle multidimensional physics from the point of view of general relativity, rather than particle physics. However, i guess that a word of caution is in order: i’ve read very little about these (to me) novel approaches to KK theories, and i’m not yet ready to endorse them; if they were right (and i definitely wish they were), they’d be quite revolutionary: for instance, they explain quantum indeterminacy as a result of particles travelling in higher dimensions… that’d be extremely cool (and actually make real one of my silly ideas of old), but perhaps too cool to be true? Well, i’ll leave it for you to decide (as for me, i think i’m going to read Wessan’s book, Five Dimensional Physics, lest student dreams can really come true!).

Returning to Bertolami’s paper, let me mention that it is part of a forthcoming book entitled Relativity and the Dimensionality of the World, the good news being that the above link points to freely available versions of many of its chapters, written by various authors, including Wessan and G.F.R. Ellis. The latter writes about his rather original ideas on time in General Relativity, and the Block Universe idea, familiar to all relativists, of a world represented as a frozen 4-dimensional whole. Ellis observes that such a representation clearly suggests that time is an illusion: the entire universe just is. The problem is that such a view seems incompatible with irreversible, macroscopic phenomena, as well as with the fundamental indeterminism inherent to quantum mechanics. To take into account these facts of life, Ellis proposes an Evolving Block Universe: time passes; the past is fixed and immutable, and hence has a completely different status than the future, which is still undetermined and open to influence; the kinds of `existence’ they represent are quite different: the future only exists as a potentiality rather than an actuality. The point being that our regular, predictable universe models are based on too simplistic assumptions and oversimplified systems, and that taking into account realistic, emergent ones renders the future under-determined. Although very interesting from a philosophical point of view, Ellis ideas need much fleshing out before becoming a solid theory of anything. But still, he makes many a fine point, and quite a lot of good questions worth thinking about.

A digression: Hertz’s mechanics
Finally, Bertolami’s paper draw my attention to a facet of Heinrich Hertz‘s work i was totally unaware of, namely, his contributions to the interpretation of classical mechanics. After gaining a place in the history of physics with his experimental confirmation of the existence of electromagnetic waves, and before his tragic death when he was only 37, Hertz wrote a book, The Principles of Mechanics Presented in a New Form, where he proposed a formulation of Newtonian physics freed of forces, using instead a variational principle. According to Hertz’s principle, particles move along paths of least curvature, where the (three dimensional) metric is defined by constraints instead of forces. Similar principles were proposed by Gauss and d’Alembert before Hertz, but the latter was notorious (if only ephemerally) for pushing to the forefront a view of space-time defined by matter in a purely relational, Leibnizian fashion: Hertz tries to derive his system of the world from material particles alone. Unfortunately, i’ve found little information on-line on Hertz’s ideas, which seem to be better known to philosophers due to their influence on Wittgenstein (who directly mentions Hertz in his Tractatus). For those of you with a philosophical soft spot, this paper presents a re-interpretation of some of Wittgenstein’s ideas under a Hertzian perspective. As a physicist, i find Hertz’s ideas interesting almost only as a historical curiosity, and don’t know how relevant they really are to modern epistemology: comments welcome! ;)

Nature’s nifty tricks

July 10, 2006

Herman BondiA few weeks ago, and thanks to this post over at A Neighborghood of Infinity, i discovered a little jewel: Herman Bondi‘s book Relativity and Common Sense, an insightful and original introduction to special relativity for the layman. The above mentioned post gives an overview of Bondi’s approach, based on what is known as k-calculus, arguably simple enough to be taught at highschools. Bondi makes the case for an understanding of SR as an evolution of Newtonian ideas, rather than as the revolution we all thought it was. Whether successful or not, this attempt leads Bondi to start his book with a delicious review of classical physics, illustrating the concepts and principles with very well chosen common life phenomena and their corresponding explanations. You know, trying to come to grips with quantum mechanics interpretations or finding a unified theory of everything is all very well, but it’s the marvel of being able to explain those other nature’s nifty tricks what draw me to physics in the first place. These little and easy to understand principles convey in a very real and fun sense the magic of ours world’s understandability. Bondi touches a lot of these magic things in the first fifty pages of his book, and i thought some of you may amuse yourselves finding (or just remembering) the explanations to these down-to-earth phenomena:

  • Conservation of momentum. Your baby is sleeping in her pram. Would you put the brakes on or off to ensure she will still be there when you come back and find that she awakened in the meantime?
  • Spinning cats. Can you think of a way of exploiting angular momentum conservation to explain how a cat manages to land always on all fours? (For extra score, how’s this problem related to Yang-Mills theory?)
  • Coriolis force. Can you explain it in simple terms (involving, say, a disk and a little ball)? Does it have anything to do with kitchen sinks? And what about weather? How does it explain cyclones and anticyclones?
  • Wave phenomena: the sonic boom. Can you explain how the explosion associated with a plane travelling at ultrasonic velocities is produced?
  • More on waves: Doppler effect. You surely know about the proverbial band on a train wagon, but what about putting the band on the station and yourself in the train? Will be the frequency shift identical? Why or why not?

These look like simple, even basic, questions, but their solution does not lack subtle points. I’m with Bondi in that one should better understand everyday physics before jumping to the unification of forces or even, more modestly, special relativity. At least in my case, and with the benefit of hindsight, i regret having jumped too early into mathematical physics and abstract stuff, only later learning about funny things like those above: i’m sure i’d be a better physicist had i spent more time in the wonderful world of understandable experiments… but, oh well, you already know the line: you’re never too old… ;-)

Turning back to Bondi’s book, never mind some of the unfavourable comments at Amazon. Even if you don’t buy his claims about SR being common sense (i don’t), the book is just excellent.

Parenthetical geometry

June 25, 2006

sicmIf i were asked for my preferred field in physics, i’d have a really hard time choosing just one, but it would surely relate in one way or the other to differential geometry. Picking up a programming language, on the other hand, would be far easier, for i feel at home among Scheme’s lots of infuriating, silly parenthesis. Jack Wisdom and Gerry Sussman have managed to bring together the best of both worlds in their book Structure and Interpretation of Classical Mechanics, freely available on-line. The book, aptly dedicated to the Principle of Least Action, is an amazing journey through modern classical mechanics, using two apparently different languages: differential geometry and scheme. There you’ll find all the expected topics: Lagrangian and Hamiltonian formulations, the rigid body, phase space structure, canonical transformations and a very complete treatment of perturbation theory in non-linear systems. What makes this book different (and extremely fun) is what one may call its computational stance: in the authors’ words,

Computational algorithms are used to communicate precisely some of the methods used in the analysis of dynamical phenomena. Expressing the methods of variational mechanics in a computer language forces them to be unambiguous and computationally effective. Computation requires us to be precise about the representation of mechanical and geometric notions as computational objects and permits us to represent explicitly the algorithms for manipulating these objects. Also, once formalized as a procedure, a mathematical idea becomes a tool that can be used directly to compute results.[…]
Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning.

As i mentioned, the computer language chosen is Scheme, for pretty good reasons (besides the obvious one of Sussman being one of the language’s inventors). To begin with, Scheme’s syntax is so simple that one can learn it on the go, although that simplicity does not preclude in any way powerful abstraction means and natural expression of symbolic computations. As a matter of fact, it favours it, as testified by the book’s accompanying library, scmutils. Thanks to it,
expressing mathematical equations in a way understandable by a computer is often a natural exercise. On the other hand, Scheme is an interpreted language, which means that you have at your disposal an interactive environment to play with. The convenience of it for exploratory purposes is hard to overstate.

Wisdom and Sussman have been using SICM for teaching classical mechanics at MIT during several years, and you can find additional material in the course’s website. But the fun does not end there. The 2005 booklet Functional Differential Geometry is an unconventional introduction to differential geometry using SICM’s schemy approach, which is also being followed and extended to Lie Groups by Will Farr, who has started an effort to port (part of) scmutils to PowerPC architectures [1]. Finally, there’s a SICM reading group in sore need of contributions and discussion (hint, hint), but with some potentially useful tips.

Happy hacking!


[1] Those of you who enjoyed my swimming post will probably be interested in this article by Wisdom that i found in Will’s blog,

The cyclist team

May 29, 2006

 Fieldtheory Figs FeynmancovGeorgia Tehch’s Pedrag Cvitanovic’ and friends write physics and maths books under the nome de guerre of The cyclist team. These books are interesting in many ways. First of all, they are comprehensive and of excellent quality, although, fortunately, these are not extremely rare in the field. What is not so common is the dose of humour and engaging wittiness distilled in their pages. And, besides, they’re being written over a span of many years in a totally public manner: you can view and download them in PDF of Postscript and are frequently updated. As they explain in their webbook rationale, they don’t even plan to ever publish them:

The relevant parts of a good text will be printed and perused, no less than a good electronic preprint. A bad text should be junked anyway. If a student in Buenos Aires or Salamanca reads a chapter and is wiser for it, that is all it takes to make us happy. The webbook has done something to further little piece of wisdom that we know and love.

The oldest book is Pedrag’s (Quantum) Field Theory, and its companion (and more modern) lecture notes: Quantum Field Theory, a cyclist tour. In Pedrag’s own words:

Relax by reading Classics Illustrated, diagrammatic, Predragian vision of field theory. The exposition assumes no prior knowledge of anything (other than Taylor expansion of an exponential, taking derivatives, and inate knack for doodling). The techniques covered apply to QFT, Stat Mech and stochastic processes.

As is a norm, the book’s site contains many other bits of additional information, including the delicious fable of Quefithe.

Next comes the Group Theory Book, which, under the subtitle of Birdtracks, Lie’s and Exceptional Groups and spanning almost 300 pages, will tell you all you’ll ever need about Lie Groups and Algebras. This nice PDF presentation makes for a good summary of its contents, or, as Pedrag says, of “most of the Webbook at a cyclist pace, in 50 overheads” (see also here for more short intros). In case you’re wondering, birdtracks are to Lie Groups what Feynman’s diagrams to QFT, and then more. As you can see, Pedrag loves diagrams and pictures, in a way that reminds me of Penrose’s fondness for geometrical descriptions (actually, birdtracks have many a point in common with Penrose’s diagrammatic tensor notation, who even wrote a letter claiming his precedence on it). And, again, don’t miss the book’s site for lots of additional goodies.

Finally, there’s the Chaos Book, probably my favourite. Again, the authors introduce it far better than i would:

Quite a few excellent mathematics monographs on nonlinear dynamics and ergodic theories have been published in last three decades. On the whole, they are unreadable for non-mathematicians, and they give no hint that the theory is applicable to problems of physics, chemistry and other sciences.
By now, there are also many physics textbooks on “chaos”. Most lack depth, and many of them are plain bad, emphasizing pictorial and computer-graphics aspects of dynamics and short changing the student on the theory. That’s a pity, as the subject in its beauty and intellectual depth ranks alongside statistical mechanics and quantum field theory, with which it shares many fundamental techniques. The book represents authors’ attempt to formulate the subject as one of the basic cornerstones of the advanced graduate physics curriculum of future.

The amount of additional information for this book is almost overwhelming, including computer programs, additional exercises (the book itself contains many) and a long list of projects written by students. I won’t try to summarize the wide range of themes covered by the book (here you have the table of contents of its three volumes–classical chaos, quantum chaos and appendices), but a very good way of getting a glimpse of its scope and fun style is reading its Overture (PS). An amazing way to become acquainted with an amazing subject!

The cyclist team

Hooke manuscript is returned home

May 18, 2006

BBC News is reporting about a lost and found Hooke manuscript (see also this older news):

The hand-written notes are thought to contain a “treasure trove” of information about the early endeavours of the UK’s academy of science. The document, which had lain hidden in a house in Hampshire, was rescued from a public auction after a fundraising effort pulled in the £940,000 needed.

As you probably know, Hooke was famous (among other things) for his frequent disputes with other physicists (most notably Newton and Huygens) on the priority of various discoveries, and this new manuscript could help in settling (a bit late) some of them.

Bootstrapping

May 16, 2006

As we all know, one cannot pull oneself up by the bootstraps, except for Baron Münchhausen, who is told to have lifted himself out of a swamp by the simple device of pulling himself up by his own hair. As for myself, i’ve pulled my hair lots of times to no avail and, as a result, i have some empirical reasons to question the Baron’s deeds!. You probably know the that the reason for our boostrapping inability is, basically, Newton’s Third Law, from which it is easily derived that the center of mass of a set of bodies can only accelerate as the result of external forces. Thus, no matter how much wriggling, you won’t be able to swim in empty space. Or will you?

Center of mass in a sphereWell, actually there’s a twist: you won’t be able to swim in empty, flat space. As it happens, curved space gives you rope enough to boostrap yourself! This well known phenomenon is explored in detail in the delicious Swimming in curved space or the Baron and the cat, recently published by the New Journal of Physics (which, by the way, is an open-access publication, although it maybe falls a bit short of what every scientific journal should be). The authors provide a detailed account of how Riemannian curvature can be used as a sort of handle to pull you around. Even if you don’t feel like following all the mathematical details, you can see movies of the swimming in action over a sphere or a hyperboloid, showing how the periodic motion of three particles on constant curvature spaces gives rise to a net displacement of their center of mass (which carries the particles around in the process). As the article explains, the gist of this effect hinges on the fact that the center of mass is only well-defined in Euclidean space: if you take for instance three equidistant (yellow) points on a sphere, their center of mass would be located at the red point, but if you move the two bodies on the right to the red dot, the new ‘center of mass’ is displaced to the blue dot.

Swimming in a circleThere are other simple examples of boostrapping. For instance, in spaces with non-trivial topologies like a circle. In there, a particle (red) can split in two parts (blue) that recombine in the opposite side (hollow red) without violating Newton’s law. A similar effect is possible in spaces with intersecting geodesics, as i’m sure you can convince yourself with a little thought (or just cheating and looking at the article’s appendices). Note also that this fun magic works on the basis of geometry alone: there’s no need of relativity or anything, just plain Newtonian mechanics.

Over at MathTrek, Ivars Peterson is talking about Edwin Abbot’s classic Flatland. Taking into account how fun this little book is as it is, just imagine how much more interesting could the history have been had flatlanders lived in a curved two-dimensional space. Any takers? :)

Fun problem

April 23, 2006

In a recent post on sci.physics.research, Igor Khavkine proposes a fun problem (requiring no more than high school classical mechanics) whose solution, he claims, is ‘a little surprising and intriguing’. Here is the problem:

BallConsider a point particle sliding on a flat table (ignore friction). The table has a cylindrical hole of finite depth (vertical walls, flat bottom). The particle can approach the hole with different velocities and with different impact parameters (the particle’s motion need not be directed toward the center of the hole). As the particle falls into the hole, it starts bouncing off the walls and the bottom (assume elastic collisions). Sometimes it gets stuck in the hole forever, sometimes it escapes (bounces out). Determine the relation between the depth of the hole, its radius, the particle’s initial velocity, and impact parameter necessary for the particle to escape after it falls in.

Some people have already chimed in with comments, but you may want to try your hand at it first!

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The Chymistry of Isaac Newton

April 9, 2006

When i was young(er), i had the immense fortune of having a mentor, Dr. Fidel Antonio Alsina, who guided me into discovering the wonderful world of physics. His lessons (by letter at first and later, when he came to Barcelona, in person) were mostly centered on relativity, but there were lots of opportunities to talk about physics and epistemology as well. He taught me to keep an open and critic mind, and not to accept arguments on the grounds of authority alone: he was fond of remembering that Newton believed sunspots were windows used by the inhabitants of the Sun. But, at the same time, he warned me against becoming a crackpot (“sages are silly, but not that silly”), and to study hard and respectfully the writings of great physicists, like Newton himself. I remember Fidel recommended me reading Newton’s Queries at the end of his Optiks to get a glimpse of how Newton’s mind worked, an advice that i can’t but pass on to all of you with a serious interest in science. As it comes, i can do even better, and point you directly to the Questiones quaedam Philosophiae, which are available online as part of The Newton Project, an ambitious initiative to put online all of Newton’s manuscripts. A related project is The Chymistry of Isaac Newton, which is publishing Newton’s alchemical works. I’m not specially interested in the later, except for their publication of Newton’s most complete laboratory notebook, which includes many of his optics experiments. A great way of seeing a great mind in action.

If you have not yet read any biography of Newton, of course the canonical reference is Westfall’s awesome Never at Rest : A Biography of Isaac Newton, but let me also recommend (for those of you with less time in your hands, and a library at hand, since it seems to be out of press) Gale E. Christianson’s In the presence of the Creator: Isaac Newton and his times, a book i enjoyed immensely.