I would strongly suggest you learn Lagrangian and Hamiltonian Mechanics from this book first  since it comes with many more illustrations and simple arguments that'll make reading SICM much easier. If you don't have time to read a whole book and want to get the main idea I've written a blog post about Lagragian mechanics myself  which has made it to the front page of Hacker News before. The great thing about SICM is that it's a physics textbook where the formulas are replaced by code  which you means you can play around with your assumptions to gain intuition for how everything works.
IMO I believe in introductory physics we overemphasize formalism over intuition and playing around with simulators is a truer way to explore physics since most physical laws were derived via experimentation not derivation. Another book that really drives this point home is 
The path taken by the system between times t1 and t2 and configurations q1 and q2 is the one for which the action is stationary (no change) to first order.
The reason for this is quantum mechanics
This is incorrect. You dont need QM to formulate, derive or use the LAP. This makes even less sense in the context of the book.
> The QM phase answer provides a deep explanation for why least action occurs at a classical level.
No, it does not. The phase in a QM state provides the intereference of the probabilities, which is an integral part in the calculations on the many-paths formulation of QM, it has NOTHING to do in the classical sense.If that is true, please derive the GR action from QM, if you do so a Nobel prize and a seat along Newton and Einstein are waiting for you.
> QM and Classical are far from equivalent. QM looks like classical under many macro situations.
These two statements are contradictory.Maybe you are misremembering the Ehrenfest theorem. If that is the case you are confusing the expectation value of a physical quantity in QM with an actual physical measurement.
On the matter of automatic differentiation, if you check out the scmutils source code, there's been an ongoing effort spanning ~a decade to fix a very subtle bug...
But honestly the book is very cheap relative to how good it is IMO
If someone is looking for the PDF format of SICM, that's available now too for the second edition.
> When we started we expected that using this approach to formulate mechanics would be easy. We quickly learned that many things we thought we understood we did not in fact understand. Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning. The resulting struggle to make the mathematics precise, yet clear and computationally effective, lasted far longer than we anticipated. We learned a great deal about both mechanics and computation by this process. We hope others, especially our competitors, will adopt these methods, which enhance understanding while slowing research.
This second edition is from 2015, following the 2001 first edition.
Unfortunately, unlike SICP, it does not seem to be under a free license, so it is not legal to translate it into Spanish or produce a reformatted digital version that incorporates an actual Scheme interpreter.
[Edited to add NC.]
The two small warnings I would share with someone starting this book are
1) they introduce some of their own notation to clarify, i.e., what various derivatives mean, but this notation is different than what is found in other texts
2) it ramps up pretty quickly from solving a double pendulum to much higher-level stuff like Lie transforms and perturbation theory - it's a lot to keep in your head all at once. Don't get discouraged if you hit a wall and need to come a couple days or weeks or months later - I definitely did, and it is still fun to try to go back and make it through the harder parts. Highly recommend!
OK, so here's my question: the scheme is great and all, but wouldn't this really benefit from a statically typed language with a rich type system? I think it would be really interesting to try to make the computations at the type level correct as well as at the runtime level. Obvious candidates I guess are Haskell/OCaml etc, or one of the theorem-proving languages (out of my depth here, but Lean/Coq etc).
I have said this before...another HN thread on SICM: https://news.ycombinator.com/item?id=21460106
I don't think the language needs to be statically-typed to do that. I could easily extend Scheme or Python to support that. All you need to do is make sure that if you add 5 meters to 5 seconds, you raise an exception. There's no reason that exception has to happen at compile-time.
Going Scheme -> typed Scheme is a manageable step which I could envision taking a few weeks of hard work. Changing languages would be an incredible amount of work (especially with things like the JIT compiling MIT-Scheme does for SICM, where you can symbolically derive equations of motion, compile them, and run them as high-performance native code).
I've been using programming to teach kids math and physics, and the lack of units IS a serious pedagogical problem. I think that's doubly true here.
I think you're really onto something.
I have an educational resource for introduction to Hamilton's stationary action. The title is "Least action visualized".
The diagrams on the page have a slider for active exploration. Moving the slider sweeps out a range of trial trajectories. As you change the trial trajectory the diagram shows how the graphs of the energies come out accordingly.
In this resource Hamilton's stationary action is introduced in a two-stage process.
We have the Work-Energy theorem, which we can apply with equal validity in infinitisimal form. The true trajectory has the following obvious property: at every instant in time the rate of change of potential energy matches the rate of change of kinetic energy. Demanding this match as a condition we identify the true trajectory among the range of trial trajectories. That is, this initial stage is already variational approach, but it doesn't yet use the concept of action.
Demonstration of moving in a single step from the first stage to Hamilton's stationary action.
The demonstration is for the simplest case: a uniform force, hence a linear potential. The reasoning generalizes to all cases.
Not so incredible; Gerry has always loved physics and in the 80s built a digital orrery in scheme shortly after SCIP was written. I believe this book had its earliest inklings in that work.