
Lagrangians and Hamiltonians for High School Students - kqr2
http://arxiv.org/abs/physics/0004029
======
ewjordan
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.

~~~
kurtosis
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!

~~~
seanc
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?

------
michael_nielsen
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...](http://www.scribd.com/doc/8321714/Vol-2-Ch-19-Principle-of-Least-
Action)

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

~~~
rms
Linking is not a crime.

~~~
kirubakaran
... if you are not in Australia.

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

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seanc
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"?

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

~~~
seanc
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...

~~~
lincolnq
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.

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hs
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?

~~~
kurtosis
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.

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

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

