Help in writing for high school/undergrad physics

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I'm writing some pieces with novel approaches on basic orbital mechanics and angular momentum in Euclidean geometries, Einstein's special theory of relativity, centripetal force/acceleration, and deriving equations for pendulum motion. They are worked out, but not completely written, and I'd very much appreciate any thoughts someone might provide. Two require only algebra and two require at least some of 1st year calculus. If there is any interest, I could certainly use the help and I'll try and post something to read.

Jon

Um, just in case I wasn't clear, I'm not asking for anyone to write things for me or do any creative work. I will do all the drawings, lay out the flow of equations, write the text. I'm just looking for criticisms from folks who are imaginative and have good grounding in algebra and a little bit of calculus (the pendulum part is the toughest that way since it requires some ability in 2nd order, ordinary differential equations... but I can finesse across some of that.)

Jon

Okay. Now I'm worried no one here does much of any math or physics, routinely. Perhaps some WP classes are in store!! I need victims... I mean students!

Jon

I'd be glad to look it over real quick if you want. I've taken a lot of university level maths so no problem there. But I probably can't do more than just some quick comments.

Every little bit would be appreciated. Are all the subjects I mentioned equally boring? Or is there any one you might be more interested in seeing than another?

Jon

I NEVER venture into this section but I have a better question than Jon has, It's a question even Jon cant answer and it's what ALL higher math problems look like me. Riddle me this, Batman!

If it takes a man half an hour to dig a half a hole with a shovel, how long does it take for a one-legged grasshopper to dig the juice out of a dill pickel?

Um... thanks Beck for the interesting math question. The answer is "Just long enough!"

So riddle me this, pesky besky:

love you,
Jon

It sounds like it might be relaxing, after wrapping my head around "The Road to Reality", recently.

Penrose, I see. I haven't read it. Does it include a fair amount of math in it?

One of the few more favorite biographies of physicists is Jagdish Mehra's "The Beat of a Different Drum: The Life and Science of Richard Feynman." It not only includes great stories in it about his life and childhood, but it is filled with the development of quantum electrodynamics and pretty much all the necessary equations and developments, in between. To read it well, you need to know math, though. But it is probably the best kind of biography a physicist could hope to have -- to include all of how they saw the world around them. In Feynman's case, his viewpoint experienced a great deal of trouble because Feynman didn't bother with developing his ideas into traditional equation form, but instead in a much more visual/conceptual way. And until people such as Freeman Dyson stuck like glue to him for a while and helped translate his way of thinking into more traditional means of providing quantitative detail, Feynman was ostricized quite a lot. So the book covers all sides and does it pretty well, I think.

Do you have any preference on what I start with?

The basic orbital mechanics and angular momentum in Euclidean geometries thing is mostly an infinitesimal approach combined with geometric visuals, but the conclusion is interesting in showing how the concept of angular momentum can arrive as a necessary deduction from the assumed geometry itself. That's a little newish, I think, because angular momentum is usually taught as a "thing" that people just accept as a law, without really being able to see why it is inescapable. Also, that piece also shows how no matter what sign the equation of gravity might have taken, repulsive, neutral and not present, or attractive and regardless of the specific equation itself, the concept of equal areas in equal times is really just an initial conditions thing and wouldn't change in the least. These things make the approach I want to take different from those I've seen elsewhere.

The development of Einstein's special theory of relativity is well known and I don't plan on changing any of that -- it's a very simple algebraic process, no calculus involved, and flows out of one very interesting assumption. The point of this one is to call attention to that assumption, more closely. I'd develop the equations, of course, but I'd then refocus the attention at the two starting and very simple equations which are simply two expressions of the same equation with different variables, really. Anyway, the interesting part of this one is the starting assumptions and asking what they may "mean" and imply. I might also suggest a thought towards the general theory of relativity, from there.

The reason I wanted to focus on centripetal force/acceleration is that the high school physics books treatments I've read are far too complex and far too little in terms of visualization of the problem. In fact, I've an entirely _new_ approach to "seeing" the problem, that I've never seen anywhere and I've been asking other physicists to tell me if they've seen it elsewhere without anyone saying that they have. It provides an interesting new tool to use in considering others kinds of complex motion, which actually leads into the 4th thing I wanted to talk about, pendulum motion.

The pendulum motion paper will develop the idea of using infinitesimals instead of more traditional calculus methods, to ease into the derivation of the usual pendulum equation (which is really just a narrowing of the general case to help simplify the equation solution.) The main thing is to highlight the use of infinitesimals, but I also do that in the orbital mechanics paper, so this one acts in support of that one but in a slightly more complex problem of constrained motion. I will also tie in the visualization tool of the centripetal force/acceleration paper, so it is probably better that this one follows that one in terms of reading through it.

So, any preferences from anyone (including you?)

Jon

Sorry... it turned into a busy evening.

I don't think I've ever read a biography.

The Penrose book does have two pages out of the 1,000-odd that have something other than mathematics on them, I think.

I can't follow what you mean with "Also, that piece also shows how no matter what sign the equation of gravity might have taken, repulsive, neutral and not present, or attractive and regardless of the specific equation itself, the concept of equal areas in equal times is really just an initial conditions thing and wouldn't change in the least."

I find it interesting that you consider "infinitesimals" not to be the same as "traditional calculus methods", unless by the later you mean rote formula transformations.

So, I guess I'm prepared to be surprised.

By that, I mean the original viewpoint taken by Newton or, if you look for rigor, the recent work by Abraham Robinson on non-standard analysis. It's a slightly unfamiliar way to go, but once to take to it, it is easier. The traditional methods are those of Dedekind and Weierstrass, which is the method taught almost everywhere.

I didn't expect you to immediately follow the sentence you picked out. It arrives from looking at what I'll write and you'll understand better, then.

Jon

I assume you know then (or at least have heard of) Lagrangian and Hamiltonian dynamics? It's certainly one of the primary starting places where a physicist would start if you asked them to justify from first principles, things like conservation of angular momenta or kepler's laws. It certainly is far out of reach of a high school level material, even with calculus, though.

Similarly, special relativity can be worked out (some would say more elegantly) in the same framework of differential geometry that the general theory does, working in minkowski spacetime (this interpretation actually almost predates the idea of curved spacetime in the general theory)

I am primarily a maths-person so the more maths there is the more interested I am

Lagrangians are beyond where I want to go and I completely agree it is beyond high school or 1st year calculus. I need to keep it simpler. Minkowski's spacetime is pretty simple, but I honestly wasn't intending to put that final cap on things with the special theory, either.

I'm glad to see there is someone here I can learn from.

Jon

Ah, I'm just a student who's taken a lot of undergrad maths and some graduate math.

As I said...

Jon

The ads on top of WP are probably going to dramatically reduce my presence here, as if there ever was much of one. But I'll still finish up with at least some of what I wanted to cover here. Thanks in advance for any comments, too!

Jon