Tarun's discussion of Lagrangian mechanics

Dear Autodidacts,

On Sunday, February 3, at 4 pm, Tarun will lead us in a discussion of chapter 2 of Herbert Goldstein’s “Classical Mechanics” (2nd ed.), covering the basics of Lagrangian mechanics. We covered some of this material last time, but Tarun’s coverage will likely be deeper and more thorough. (This is the discussion he was prevented from presenting in January because he had the flu.)

The February 3 meeting will be held at the Brooklyn LaunchPad, a self-described “creative gathering place” that hosts events related to art, technology, and other subjects. The LaunchPad is excited to host our group, and its location seems nearly unbeatable: a five-minute walk from the 2,3,4 & 5 trains; one stop from the Atlantic Avenue station; etc. Let’s try it this time. Going forward, we can decide whether to stick with the LaunchPad, return to Dimitri’s dental office in Astoria, or oscillate back and forth at the rate of 6 reciprocal years (i.e., every other month).

Please read (or re-read) chapter 2, and save or post here any questions you may have in advance of the meeting. Tarun will likely touch on some very basic threshold questions, in addition to doing the symbol-manipulation. For example, should we be comfortable treating q and q-dot as independent variables? And what makes us think we can minimize a function over an infinite-dimensional object like a “path”? He raised these questions at a recent breakfast, and they seem completely up our alley and worth airing.

See you all on February 3. Note: This will definitely not be repeat of January's meeting -- so please don't skip in on that account. This will be our "real" coverage of the Lagrangian formalism.

- Matt

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  • Wael

    So, I'm putting down tentatively Sunday the 17th as the date that we meet to go over some problems. I find that a really good way to learn from each other is that when we get together, anyone who's solved a problem should try to get up and present the solution to the group. It's twofold: you share your insights, and you gain by learning how to organize and communicate your ideas about the physics :)

    February 4, 2013

    • Anna

      I like the idea of doing some problems...but considering that next weekend is a long weekend, would Monday the 18th work?

      February 10, 2013

  • Dyutiman D.

    didn't have time to work this out yet, but I think this could be an interesting exercise, inspired by Tarun's description historical work of Lagrange. Consider a ball falling from height h. Phase space
    (x,\dot{x}). Starting point is x = 0, \dot{x}=0, and the final point being x=h, \dot{x}=\sqrt{2gh}. Now consider a few different paths joining the two end points, one being a straight line \dot{x}=\alpha x, \dot{x}=\beta x^{2} with appropriate \alpha and \beta. Also consider the Newtonian solution of \dot{x}=\sqrt{2gx}. What's the relationship between (time) average of T and V along each path? Which path has the minimum action? Now if only I can develop some picturous intuition about this!!

    February 4, 2013

    • Wael

      Minimum action would be the Newtonian path.. because minimum action is supposed to describe the physical path (this being Hamilton's principle)

      February 5, 2013

    • Dyutiman D.

      of course, but I think it would be interesting to verify it.

      February 6, 2013

  • Matt S.

    ... one for each variable. Now, if you have some other coordinate system, say, spherical polar coords or whatever, then any path deviation you take in the "theta direction," say, can be broken down, at each point, into a little deviation in the x direction, a little one in the y direction, and a little one in the z direction. I.e., the small continuous change of path in the theta direction can be resolved into a sum of three changes of path in the x, y, and z direction. These paths in x,y,z directions will be shaped differently, since the change of variables may not be linear. But, so long as they are normal relatively smooth functions -- they too will be small deviations in the x,y, and z directions, and the E/L equations say that it doesn't matter what shape they are. So long as they are fixed at the endpoints, the action is still at a minimum. Therefore, the action of the theta deviation -- the one that reflects the sum of the x, y, and z deviations -- is also at a minimum.

    February 4, 2013

  • Matt S.

    Two thing: (1) All who were at the meeting, please let me know if there were aspects you thought could be done better. I think it was good that we got back on track and went with the basic derivation of the Euler/Lagrange equations, and I would like to push in that direction for all these "theory meetings" -- i.e., sticking close to the assigned reading, so that people know what will be discussed. Then maybe some time for special topics at the end. (2) I think I got some intuition for why the E/L equations are independent of coordinates. If you are on a minimum path, it means that any small perturbation of the path in the x direction, y direction, and z direction will lead to an E/L equation for each of the three variables. I.e., since any small deviation at all must lead to zero change in the action to first order, certainly any small deviation confined to the x direction, the y direction, or the z direction must independently meet that criterion. So you get three EL equations,

    February 4, 2013

  • Tarun

    On my way! Got lost on bike, sorry about the delay

    February 3, 2013

    • Matt S.

      Great job, Tarun. Thanks for doing this. Hopefully we'll get someone else to run the next one. Likely not a working research physicist -- but no matter, whoever does it will learn a lot in preparing for it.

      February 3, 2013

  • A former member
    A former member

    I'm sick. I forgot to mention it last night. Hope it went well, see you wonderful people next month.

    February 3, 2013

    • Matt S.

      Pedro, the meeting wend very well, and it seems likely we're going to have a more-or-less permanent affiliation with the Launchpad. So things are looking up. Hope you feel better and keep the first Sunday of the month clear until further notice. Matt

      February 3, 2013

  • Tarun

    If time permits and people are interested, I can also speak a little bit about a big controversy that is brewing in high-energy physics — No one knows the real radius of the proton! [0] Basically, someone did a new experiment that finds a significantly different proton radius. If we believe the old experiments to be true and we assume that we have done enough independent experiments to apply the central limit theorem and estimate a probability distribution over radii, the radius found from this new experiment would be a 7-sigma event. While this is a quantum phenomena [1], I will try to draw some classical analogies to explain why this result is controversial. It reminds you that even within the astonishing precision of high-energy physics experiments, there is room for much error. [0] http://www.sciencemag.org/conte...­
    [1] More specifically, its a problem with the exact description of the strong force quantum chromodynamics.

    February 2, 2013

    • Tarun

      Correction to the first footnote: More specifically, its a problem with the exact description of the interaction between the strong force (described by quantum chromodynamics) and the EM force (described by quantum electrodynamics).

      February 2, 2013

  • Tarun

    I have a presentation that I gave for a small group of non-physicist researchers (with a math background) on symmetries and Lagrangians. Here's a link: http://www.tarunchitra.com/cond...­

    January 13, 2013

    • Matt S.

      Tarun - Are you good for Sunday? I realized it's Superbowl Sunday. It looks like we're going to have a good turnout anyway. You don't need to make it a "lecture" -- just be ready to walk through the steps of the derivations, and kind of field questions, which you can then throw out to the group if you don't know the answers. - Matt

      January 31, 2013

    • Tarun

      Yep that's totally fine!

      February 1, 2013

  • Matt S.

    In going from equation 2-7 to 2-8, Goldstein first uses the fact that you can take mixed partial derivatives in either order; he then switches the order of differentiation, and then integrates by parts over the x variable. In integrating over a partial derivative as if it were a total derivative (i.e., integrating "dx"), Goldstein confused me. I now think I understand why he could switch from a partial to a total derivative like this. (He's keeping alpha as a parameter in the equation, and he's just integrating over the x variability alone.) But, I could use some more insight to make sure I'm getting this straight.

    By the way, to see the picture of why mixed partials can be taken in either order, see this: http://www.physicsinsights.org/...­. I really like picture proofs like this.

    January 13, 2013

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