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On a personal note, I consider the university class where I learned about Lagrangians for the first time to be a pivotal moment in my life. It was the first tim
by qchris 7y ago
On a personal note, I consider the university class where I learned about Lagrangians for the first time to be a pivotal moment in my life. It was the first time I experienced a fundamental sense of awe and beauty about a mathematical construct, in physics or otherwise. I'd thought physics was useful and interesting before that, but seeing derivation of and application the Lagrangian formulation to mechanical systems was, and is, simply gorgeous to me. And as someone who recently began dipping their toe into neural networks, this is really very cool.
- sgrey 7y agoChris, glad you like it. I can relate to that feeling, which I also had as an undergrad. It was a big motivation for this work -- and I think the analogy can go even further. For example, there's a really cool lecun paper about NN training dynamics actually being a lagrangian system: http://yann.lecun.com/exdb/publis/pdf/lecun-88.pdf http://yann.lecun.com/exdb/publis/pdf/lecun-88.pdf. Another thing I want to try is literally write down a learning problem where S (the action) is the cost function and then optimize it...
- qchris 7y agoThat does seem interesting! After skimming that paper, I think I'm going to need to sit down with it in order to really parse through things, though. Some of the operator combinations seem to be things I haven't worked with jointly before. I'd definitely be interested to see the results of using the action as the cost function, though!
- madhadron 7y agoI kind of went the other way. I skipped out on my undergrad classical mechanics class and did the graduate course, taught by a guy from the Landau school, out of volume 1 of Landau & Lifshitz. So in the summer after my 3rd year I spent some quality time with Newton's Principia and then most of the rest of the summer staring off into space as I hooked up everything between the two systems again. Man, that was a great summer.
- s9w 7y agoOh god Landau & Lifshitz, that wakes memories. That was hardcore
- mikorym 7y agoAn economics PhD once asked me for help with Langangians, but at the time I was quite occupied with other branches of mathematics. I wonder if it would make sense to revisit it. If I recall, the problem he was looking at was quite standard, and involved perturbations (indeed, like the corona virus) to economical models.
- whatshisface 7y agoA "perturbation" is usually short for a "small perturbation," which SARS-CoV-2 is definitely not. ;)
- mikorym 7y agoMaybe usually, but iirc in his case he specifically said "major" events and recourse to normal after that.
- seisvelas 7y agoWow, that's really inspiring. I also love beauty. Can anyone recommend a way for someone with undergrad level math to learn about Lagrangians?
- rrss 7y agoThere a lot of good resources - maybe try these to start: https://www.feynmanlectures.caltech.edu/II_19.html https://www.feynmanlectures.caltech.edu/II_19.html http://www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf http://www.damtp.cam.ac.uk/user/tong/dynamics/two.pdf https://theoreticalminimum.com/courses/classical-mechanics/2011/fall/lecture-3 https://theoreticalminimum.com/courses/classical-mechanics/2...
- whatshisface 7y agoWhat's so great about it? I had that class too and I didn't think it was so hot. Maybe I missed something.
- pas 7y agoElegantly deriving the optimal path, which also happens to be what reality usually does? (Because many problems are "easily" transformed into energy minimization problems [principle of least action].) Conservation laws (and/or the associated symmetries) might be more fascinating, but that all builds on Lagrangian mechanics.
- whatshisface 7y agoWhy would you say that reality "minimizes action" when you could just as easily say it "solves differential equations?" They are two equivalent ways of looking at the same thing, and you can't really tell which one nature does.
- rrss 7y ago"solves differential equations" doesn't tell you anything about which differential equations. "minimizes action" gives you a way to get the differential equations corresponding to pretty much every scenario.
- whatshisface 7y agoSpecifying the action is equivalent to specifying the differential equations. Knowing that you start with a Lagrangian doesn't tell you more than knowing you start with some differential equations.
- deleted 7y ago[deleted]
- Brain_Thief 7y agoBut are the statements "reality minimizes action" and "reality solves differential equations" actually, literally equivalent to each other? It strikes me that this is not in fact the case. There are solutions to systems describing skyscraper sway that have the building stand at a terrifying lean angle; what is supposed to be selecting the particular solutions that we observe? A purely random walk through the solution space seems unsatisfying as an answer.