Lagrangians and Hamiltonians for High School Students | Hacker News Reader

Lagrangians and Hamiltonians for High School Students

arxiv.org

27 pointskqr215 comments

Okay, but what's the point? Yes, it's possible to write down the Hamiltonian and Lagrangian formulations in a form that high school kids can understand. To what end? The reason these formulations are preferred is that certain classes of problems and generalizations become easy when you look at mechanics from these points of view, whereas in the "usual" setup there's no obvious way to proceed, but most of these problems are too difficult for high school kids to touch, and the utility of the abstractions will be lost on them altogether. They just become another set of equations to memorize, with no idea why anyone would write them that way...

This is particularly true with the Lagrangian and Hamiltonian formulations - as presented, they are absolutely trivial rearrangements of the equations of motion. Their power only presents when you look at how they generalize to more complex situations.

It always bothers me when people try to slip advanced material into lower level education without addressing the "why?" question. A shallow explanation backed by a few equations doesn't help anyone, and in the worst cases it can make the material seem even more obscure and mysterious.

yeah this was a pretty lame introduction, but there are several BIG ideas in lagrangian and hamiltonian mechanics that I think would be entirely appropriate for HS - in fact the earlier people are exposed to these the better.

(1) The local motion always "chooses" a path which optimizes a "global" property (the action). A system which follows only local rules can find a global optimum. This is a very important idea. I think this point finally made it into a nobel lecture for economics maybe 200+ years after maupertuis, d'alembert and of course, lagrange.

(2) Knowledge of the conserved quantities completely determines the motion. Conservation laws take on their simplest form in the hamiltonian formulation. I was very fortunate to have a high school teacher who introduced me to these ideas then. It's probably the reason I decided to study physics!

Well, yes, putting the sarcasm below aside, I agree that this is Really Cool Stuff.

Reminds me of the bit in Anathem when Fraa Erasmas teaches Barb about Hilbert Spaces because, dammit, he was ready, and why make somebody go through the pain of doing it wrong when they're ready to learn how to do it the easy way?

A more interesting treatment of Lagrangians, in my opinion, is Chapter 19 of Volume 2 of the Feynman Lectures on Physics. It's a bit more complex, but I think you can get a lot more out of it. It's here: http://www.scribd.com/doc/8321714/Vol-2-Ch-19-Principle-of-L...

I'm a bit uncomfortable with posting the link to scribd, but I guess it's fair use.

Linking is not a crime.

... if you are not in Australia.

http://news.ycombinator.com/item?id=520894

Um, riiight. Are the kids doing differential calculus in high school now? Or is there another paper entitled "Ordinary Differential Equations for High School Students"?

Um, yes they are doing ODEs in high school. Not all of them, but a few.

OK, call me an old fart then. I did AP Calculus in '92, and ODE was certainly NOT on the list.

I shouldn't be surprised I guess. After all, once upon a time Special Relativity was graduate level material.

One more sign that the singularity is near, I suppose...

It's not on the AP. I did more differential equation stuff in the AP physics course that was taught in parallel with the AP calc course. Mostly simple stuff like coming across the differential equation "a = -w^2 x" and having to solve it (x = sin wt -- simple harmonic motion). We didn't often have to solve new ones on our own. This lecture would definitely be understandable to the students in that class though.

I had ODEs and integration in the last year of high school (8h/week of math).

ten years ago i studied this langrange thingy in graduate level engineering mechanics

now they teach it in high school ... wow! should i feel dumb or happy?

Happy. Baby Rudin will soon be taught to babies, and the world is better for it :-)

I think that the biggest limitation is the number of qualified teachers. Surely not everyone has the aptitude to learn these ideas, but of those who do only a tiny fraction are given access to quality instructors. It was definitely not a part of my traditional instruction. I had to stay after and nag my teacher until he explained it to me. There's not much glory in it but it's really fun to think of ways to present "advanced" topics in an elementary form.

As a lesson, this is a good idea, horribly executed.

As a sketch for preparing a lesson, it's useful.

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