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I've been thinking lately about the many things we usually leave out of intro physics. It's hard to strike the right balance between practical and inspirational, between strong foundations and broad horizons. But I'd like to see that change, at least a little: you won't entice students to love the subject without a glimpse of what makes it beautiful.
Just for fun, let me share an example: the quantum idea that reality is a sum over all possible histories. It goes something like this. When I throw a ball and you catch it, common sense and classical physics agree that the ball follows a specific path through the air.
But quantum mechanics tells a remarkably different story. In this picture, the ball takes every possible path from me to you at once: a straight line, or a high arc, or a zig-zag, or three quick loops around my head before diving between your legs and then up over your head and back down into your hand, or anything else you can imagine. Moreover (and further straining credulity), every single possibility is equally likely! Of course, with so many options, the odds of any given path are basically nil: what we really have to compare are "neighborhoods" of almost-identical paths (those similar enough that we couldn't tell them apart anyway).
Very roughly speaking, quantum mechanics says that every possible path gets to cast an equal vote in favor of its neighborhood. But there's a catch: each path gets a specific "direction" assigned to its vote (think of this like the spinner from a game of Twister). In most neighborhoods those directions go every which way, so when you add up all the votes they pretty much cancel out (just as many lefts as rights, etc.). But there's usually one special neighborhood that votes as a block, with all its vote directions more or less the same. That overwhelmingly probable neighborhood is what we see as the classical path. Common sense emerges from chaos in a truly remarkable way.
[The deeper beauty of this story is that the math of these "vote directions" immediately explains the reason behind the equations of classical physics. The vote direction of a path is precisely its "action" (interpreted as an angle). We learn in calculus that near the minimum of any function its value is nearly constant, so the neighborhood of the path with minimum action will have nearly constant vote direction. And indeed, the "principle of least action" was the basis of the Lagrangian formulation of mechanics long before quantum mechanics was ever imagined, but nobody knew why. Good stuff!]
Just for fun, let me share an example: the quantum idea that reality is a sum over all possible histories. It goes something like this. When I throw a ball and you catch it, common sense and classical physics agree that the ball follows a specific path through the air.
But quantum mechanics tells a remarkably different story. In this picture, the ball takes every possible path from me to you at once: a straight line, or a high arc, or a zig-zag, or three quick loops around my head before diving between your legs and then up over your head and back down into your hand, or anything else you can imagine. Moreover (and further straining credulity), every single possibility is equally likely! Of course, with so many options, the odds of any given path are basically nil: what we really have to compare are "neighborhoods" of almost-identical paths (those similar enough that we couldn't tell them apart anyway).
Very roughly speaking, quantum mechanics says that every possible path gets to cast an equal vote in favor of its neighborhood. But there's a catch: each path gets a specific "direction" assigned to its vote (think of this like the spinner from a game of Twister). In most neighborhoods those directions go every which way, so when you add up all the votes they pretty much cancel out (just as many lefts as rights, etc.). But there's usually one special neighborhood that votes as a block, with all its vote directions more or less the same. That overwhelmingly probable neighborhood is what we see as the classical path. Common sense emerges from chaos in a truly remarkable way.
[The deeper beauty of this story is that the math of these "vote directions" immediately explains the reason behind the equations of classical physics. The vote direction of a path is precisely its "action" (interpreted as an angle). We learn in calculus that near the minimum of any function its value is nearly constant, so the neighborhood of the path with minimum action will have nearly constant vote direction. And indeed, the "principle of least action" was the basis of the Lagrangian formulation of mechanics long before quantum mechanics was ever imagined, but nobody knew why. Good stuff!]
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Woooo, questions I don't know how to answer.
(Honestly I always thought it looked like some kind of freaky aorta. But even if I knew what it was, explaining it to a 3-year-old would be a whole other thing.)
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How's this? "Steuard's job is to figure out how all the tiniest bits of the universe fit together. That's a picture from one story about how it all might work." And if you want more detail: "The red tubes on each side are tiny pieces of stuff, and the blue tubes between them show how the pieces talk to each other."
But that probably wouldn't really get across the idea at all. Hmm. Give me a few minutes.
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(Sigh. I should go back, hit my 17-year-old self, and tell her, "Anything that feels anything like a quantum ensemble? YOU LOVE IT. Study more like that.")
All of that said, it seems to me that anything that has to do with voting/probability lends itself to a classroom simulation...
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For example, my intuition says that if I couple two pendulums together, that the "real" way to describe what's going on is to give the position and velocity of each. It's also possible (and in some ways more elegant) to talk about the eigenvectors of the system (oscillating together, oscillating opposite), and describe it as a sum of eigenvectors. My intuition may be due to how I'd interact with it: In a real system, I'll push on one pendulum or the other with my fingers, which is interacting with the position-velocity view.
I'm similarly suspicious of the principle of least action. It feels wrong to describe what happened as looking into the future to see how things will wind up, and picking some sort of optimal path. The more natural view seems to be that the time derivative of the action is always zero, and that integrating that (and looking for a minimum) is just a math trick with no particular reality.
Does that sort of question even make sense?
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In terms of your classical example of a "real" vs. "trick" description of pendulums, I feel that the two are equally "real". For many (most?) questions, the position-velocity description is the most natural, but for others the eigenstates are the most natural. Because the two descriptions are related by a simple change of basis, I don't see them as having any difference in "truth" at all. I wouldn't consider one of them to be more "real" any more than I would consider a map with north oriented up to be more "real" than a map with west-northwest oriented up. A change of basis may affect convenience, but never truth.
I'm very sympathetic to your feelings about the principle of least action: I used to feel the same way. But I've heard that this is just an artifact of the simple version of the principle that we're all usually taught. There's apparently a way to set up the action with the endpoint free (rather than specified in advance), presumably along with initial conditions for (generalized) velocity and a final time. My impression is that although the starting point would look somewhat different, the physically significant results would be the same.
And the physically significant results are just the Euler-Lagrange equations for the system. Those are the calculus-of-variations equivalent of "set the derivatives equal to zero at a minimum". The Euler-Lagrange equations provide precisely the notion of "step forward moment by moment" that you're looking for (rather than looking into the future).
[In fact, two different actions that give rise to the same Euler-Lagrange equations are equivalent. (At least classically! I've had arguments about the status of that in the quantum case in the past, and I don't think we ever quite worked out the answer.) The relative status of "Euler-Lagrange" vs. "action" feels similar in many ways to the status of "electromagnetic fields" vs. "scalar and vector potential": classically, only the first of each pair is physically significant, but there may be quantum effects that really do care about the second one. (Those quantum effects may all be topological, though. I should think more about this.)]
[For the record, your "time derivative of the action equals zero" criterion isn't really right: we should fully expect the value of the action to change over time, since we're just looking for the smallest total at the end. I think the Euler-Lagrange equations fill the mental slot you're looking for. Also, the time derivative of the action is just the Lagrangian.]
My final statement on what is "real" here has to come down to this. I've (eventually) accepted that quantum mechanics gives the "real" description of physical reality. The sum over histories formulation is my preferred formulation of quantum mechanics (for both philosophical and technical reasons), but I don't think it is more or less true than any equivalent formulation (this is essentially yet another choice of basis). And the transition from "sum over histories" to "principle of least action" to "classical equations of motion" is by far the clearest connection between classical and quantum physics that I know. With all that in mind, yes, I consider the sum over histories to be real. It's certainly "right" (to the best of our current understanding), and it's certainly "useful" or "convenient". And philosophically I don't see it as being worse than any other formulation of quantum mechanics (in fact, as I've said, I personally consider it the most satisfying).