Couple of small PSAs: > On Jul 4, 2023, at 12:28 AM, Nicholas Thompson <thompnicks...@gmail.com> > wrote: > > And why do we keep calling it by it/s cult name, rather than calling it what > it is? The difference between the energy of a moving object conveyed by its > velocity and that conveyed by its position in a field. If you want jargon, > why not just call it Ev-Ep.
It is a subset of cases in mechanics where the Lagrangian equals a kinetic energy minus a potential energy. It is a subset of those cases in which the kinetic energy is some quadratic function of velocities and some parameters. It is a subset of cases where Lagrangians and Actions even describe the variables in mechanics. The concept of Lagrangian is not defined by any one or another of these instances, and the concept of action (in the subset of cases where it is a time-integral in a single time variable of a Lagrangian) is not defined by any particular one of the indefinitely-variable functional forms that the action might evaluate to. (When read aloud in my voice, one should hear a tedious echo of harping that the concept of entropy is not defined by the functional form of an equilibrium entropy for one or another Gibbs ensemble.) If we wish to talk about the consequence of books for the course of the enlightenment, why do we use that cult name “book”, and not just pick up a copy of Moby Dick and say “the things in which the word whale appears here and here and here”? Now that Glen informs us that Sutter did mention calculus of variations, it is okay for me to just say the following and not go off on a long tangent: If I tell you that the daisy-chain recursion (aka the calculus of variations) that identifies the trajectory minimizing an action — because its (let’s say) potential term is a function of position and its (let’s say) kinetic term is a function of the rates of change of positions, so trying to reduce a function of them both chains together positions across time, and this is what “equations of motion” are — implies that the energy is conserved (unless blah blah blah, or else this sentence never ends), and that that is what the minus sign is buying you, don’t you feel infinitely better now that I have used the terms “energy” and “conserved"? Doesn’t your intuition feel infinitely more empowered, ready to go out into the world and know what’s what? I hope not. Aristotle thought about motion and we know because he wrote things down. Probably people before him thought about motion and didn’t write things down. Surely people after him thought about motion, he having made that okay to do. We waited until Newton before anybody had quantities showing that a concept “energy” was coherent and consequential. We had to wait another century-and-a-bit until Emmy Noether (using ideas from Lagrange) related energy conservation to invariance under the passage of time, so that the idea started to be anchored in some bigger context, and wasn’t just a placeholder term floating in a void. (This wasn’t what Leibniz meant by “windowless monad”, but I wish it had been; I would love to use that appellation for the placeholder status of all our terms not yet embedded in any wider contexts. (Integrated, one might say.)) I don’t say the above to mock the request for something intuitive about what least action is doing in one or another of its functional forms. If I thought I had something useful to say that fulfills the request as stated I would write it here. (The above observation about “why the minus sign” and conservation of energy is a bit in that direction, though it requires you to grind out the equations to watch it work, and in any case all this is more cleanly done by converting the Least Action formulation into its Hamiltonian equivalent and then showing how we have non-overlapping trajectories in a certain kind of picture of the space of possible conditions.) It is not at all a bad request to make, and to extrapolate up from instances is a practice without which I could never get to an understanding of anything. But I think this notion that one’s “intuition” is now empowered should be handled about like a pet rattlesnake. Or Eve’s apple. In it lies every temptation to conflate understanding with familiarity and the offloading onto habit, and if you go that route you may as well try to climb out of a black hole. Just while we’re at it: > my long-time mentor, Stephen Guerin. He has learned to be very careful > around me, but I can tell that in his own private modeling -- the kind I urge > you to tell me about above -- he believes that the lightening bolt internally > models all the possible routes to ground and then chooses the least action > path. Stephen is a generous friend indeed, to let you throw up strawman caricatures of such malice for his thinking, just for the joy he knows it gives you to throw sand into the gears of any conversation. A man like that can make a good father. Now, back to my better self which knew it should have stayed quiet, Eric > > And yet, nobody tackles the basic question. Why on earth would E in the v-Ep > be something that every moving object in the universe tries to accomplish. > > END HARRUMPH! > > > > > > > > > On Mon, Jul 3, 2023 at 2:17 AM Stephen Guerin <stephen.gue...@simtable.com > <mailto:stephen.gue...@simtable.com>> wrote: >> The Action is the integral of the Lagrangian along the whole path, not just >> a single instant. >> >> On Sun, Jul 2, 2023, 9:12 PM Nicholas Thompson <thompnicks...@gmail.com >> <mailto:thompnicks...@gmail.com>> wrote: >>> So the difference is at a positive max when the ball hits the ground and at >>> a negative maximum when the ball reaches its highest altitude? So how am I >>> to understand positive and negative? vectors? >>> >>> Instantaneious Action is at a minimum when the two terms are equal? >>> >>> I have no intuitive sense of what is going on here. >>> >>> But thanks for trying, Frank. >>> >>> N >>> >>> >>> >>> N >>> >>> On Sun, Jul 2, 2023 at 12:27 PM Nicholas Thompson <thompnicks...@gmail.com >>> <mailto:thompnicks...@gmail.com>> wrote: >>>> Frank, >>>> >>>> Thanks SO MUCH for forwarding this to me. To any other defrocked english >>>> majors on Friam, who have listened to these guys blather on about >>>> LaGrangians for all these years, I highly, HIGHLY recommend the video. >>>> Pretty short, AND, you might possibly, conceivably understand Steve Guerin >>>> when you get to the end. Yeah. Really. >>>> >>>> Nick >>>> >>>> ---------- Forwarded message --------- >>>> From: Frank Wimberly <wimber...@gmail.com <mailto:wimber...@gmail.com>> >>>> Date: Thu, Jun 29, 2023 at 2:53 PM >>>> Subject: Fwd: Watch "The Most Important Idea in Physics: The Principle of >>>> Least Action - Ask a Spaceman!" on YouTube >>>> To: Nicholas Thompson <thompnicks...@gmail.com >>>> <mailto:thompnicks...@gmail.com>> >>>> >>>> >>>> >>>> >>>> --- >>>> Frank C. Wimberly >>>> 140 Calle Ojo Feliz, >>>> Santa Fe, NM 87505 >>>> >>>> 505 670-9918 >>>> Santa Fe, NM >>>> >>>> ---------- Forwarded message --------- >>>> From: Frank Wimberly <wimber...@gmail.com <mailto:wimber...@gmail.com>> >>>> Date: Thu, Jun 29, 2023, 12:51 PM >>>> Subject: Watch "The Most Important Idea in Physics: The Principle of Least >>>> Action - Ask a Spaceman!" on YouTube >>>> To: Thompson, Nicholas <nickthomp...@earthlink.net >>>> <mailto:nickthomp...@earthlink.net>>, Barry MacKichan >>>> <barry.mackic...@mackichan.com <mailto:barry.mackic...@mackichan.com>> >>>> >>>> >>>> https://youtu.be/UuqpCBZoX3M >>>> >>>> --- >>>> Frank C. 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