Structure and Interpretation of Classical Mechanics (2014)
tgvaughan.github.io
tgvaughan.github.io
https://mitp-content-server.mit.edu/books/content/sectbyfn/b...
Scheme is pretty close to a universal computation substrate that provides enough ergonomics to be human understandable and writing anything out in it provides genuine illumination to what's going on under the hood.
The "little" books are a tease of what that series could be.
Doing the calculations automatically is a happy side effect of finding the right abstractions for describing what's happening physically and those abstractions being expressed in scheme already.
E.g. Exercise 3.73 in SCIP asks how to implement an electrical circuit using a stream data structure. Because of all the work done beforehand you end up with an expression which describes the time behaviour of the circuit using the same expressions that describe its layout.
Structure and Interpretation of Classical Mechanics (2015) - https://news.ycombinator.com/item?id=40805136 - June 2024 (12 comments)
Structure and Interpretation of Classical Mechanics - https://news.ycombinator.com/item?id=31568387 - May 2022 (1 comment)
Structure and Interpretation of Classical Mechanics - https://news.ycombinator.com/item?id=23153778 - May 2020 (40 comments)
Structure and Interpretation of Classical Mechanics (2015) - https://news.ycombinator.com/item?id=19765019 - April 2019 (87 comments)
Structure and Interpretation of Classical Mechanics - https://news.ycombinator.com/item?id=9560567 - May 2015 (20 comments)
Structure and Interpretation of Classical Mechanics - https://news.ycombinator.com/item?id=6947257 - Dec 2013 (37 comments)
Structure and Interpretation of Classical Mechanics - https://news.ycombinator.com/item?id=1581696 - Aug 2010 (20 comments)
Functional Differential Geometry (2012) [pdf] - https://news.ycombinator.com/item?id=7884551 - June 2014 (38 comments)
Of course. To him that would be modern mechanics. Or just mathematical natural philosophy, or whatever.
> Maybe we should categorize sciences based on the spatial scale at which they operate.
That would not be very useful, because there is no boundary. Nothing in general relativity says "below this everything is Newtonian". As a matter of fact we need to consider relativistic effects in quantum chemistry calculations that involve some heavy elements, at length scales smaller than 0.1 nm. Similarly, they just gave a Nobel prize for work on "Quantum properties on a human scale".
> For example, quantum scale, human scale and cosmic scales have their own physics, logic and causality.
That is not at all how these frameworks are built, and that is not the dominant epistemological approach. The mainstream view is that there is a theory of everything that exists but is unknown to us, and that our various theories are approximations of that theory under different assumptions. They look categorically different because we don’t understand the overarching framework, not because nature is fundamentally different depending on scale.
Also, I don’t see how the logic is fundamentally different between e.g. quantum mechanics and general relativity. Both rely heavily on things like Hamiltonian mechanics or symmetries. Some behaviours are different (like photons following geodesics and not straight lines, or superpositions of quantum states), but these are not a fundamental problem: a straight line is a limit case of a geodesic in a flat space, and a unique state is a limit case of superposition.
I am not saying that everything is fine and we know everything, just that there is no clear boundary between the situations in which different theories are required and we cannot neatly decompose the universe into different realms where different theories apply.
* Things don't have their own location or identity
* Spatial and temporal extents don't exist
* Something may be true and false at the same time, or concept of true and false may not be defined
* cause and effect goes for a toss, as behavior of time is different
* Existence and non-existence co-exist, or come into existence together
Similar effects at relatively-infinite scale (maybe purely mathematical)
* Comparisons (big/small/equal) breakdown
* Regular arithmetic and logic breaks down
The main thing that is at the root of all of them is the word "things". In QM, the ground truth of the world is the wavefunction of the system. The wavefunction assigns some amplitude (potentially 0) to any possible state of the system that it describes. It then evolves purely deterministically from past to future, according to Schrodinger's equation (or Dirac's equation, if you want to discuss speeds close to that of light). The only kink is interaction with a measurement device (what constitutes a measurement device is one of the big mysteries that we don't yet have an answer for). After a measurement, the wavefunction collapses non-deterministically to one of the states that the measurement device was set up to detect, with a probability that is proportional to the amplitude of the wave function of that state.
Now, this is the "ground truth" of QM. Everything else, such as particles and space-time and so on are just stories we tell to make sense of the wavefunction and its behavior. Sometimes those descriptions break down, and they start assigning weird fanciful ideas, such as retrocausality etc - but these just prove that the stories are wrong, that they are misinterpreting the math of the wavefunction.
I'd also note that the main "time is weird" factoid you encounter related to QM experiments, the delayed-choice quantum eraser, is mostly a misunderstanding / sensationalization of the actual physics and the experiment. It again only proves that certain interpretations of what the wavefunction and its collapse represent are not temporally consistent, but the direct conclusion from this should be that the interpretations are wrong, not that "cause and effect goes for a toss, as behavior of time is different".
* PNC: at most one is true; both can be false
* PEM, at least one is true; both can be true
* PNC + PEM, exactly one is true, exactly one is false
If you stick to the familiar computational complexity classes P=co-P, but NP != co-NP, and those both relate to the accessibility of T/F, where both P and NP are by definition decision problems, specifically the ones that can be verified in poly time.
If you ignore the Heisenberg uncertainty principle to avoid that complexity, the standard model of QM is just mapping continuous functions to discrete space, and will have problems with the above.
This happens in math too, where we use the rationals over the reals or Cauchy sequences to construct the reals, because almost all reals are normal and non-computable, and even equality between to "real" real numbers is undecidable.
This is also related to why after Weierstrass show that almost all continuous functions are nowhere smooth, we moved to the Epsilon-Delta definition of a limits etc...
We have the lier's paradox, which is easier to understand than Berry's paradox, which relates to Chaitin proof that there is an upper limit to what any algorithm can prove.
Things at Quantum scale do act very different than our typical intuition, but lots of maps from a continuous space to discrete categories can exhibit the same behavior even at macro scale, we just can often use a model that lets us ignore that to accomplish useful work.
Often we can even use repeated approximation or other methods to reduce those problems to something that is practical, but that is still the map and not the territory.
Superposition is just that:
f(a + b) = f(a) + f(b)
f(sa) = s*f(a)
If you think about Heisenberg uncertainty as indeterminacy instead, Independence in math (Like ZFC + CH) which are neither provable as false or true, Chatlin's complexity limits, the breakdown of Laplacian determinism, or even modal logic all apply at multiple levels.
The amazing thing in my mind is that we found useful models despite these limits and using methods which are effective.
But IMHO it is best to think that in the quantum world, we aren't so lucky rather than those limits aren't still lurking under the surface at the macro scale, which they very much are.
> That would not be very useful, because there is no boundary. Nothing in general relativity says "below this everything is Newtonian". As a matter of fact we need to consider relativistic effects in quantum chemistry calculations that involve some heavy elements, at length scales smaller than 0.1 nm. Similarly, they just gave a Nobel prize for work on "Quantum properties on a human scale".
You are just saying "well ackshually". I dare you to build a cabinet using the Hamiltonian. I double-dog-dare you.
> > For example, quantum scale, human scale and cosmic scales have their own physics, logic and causality.
> That is not at all how these frameworks are built, and that is not the dominant epistemological approach.
Again, 99.999% of all functional mechanics don't involve epistemology.
> The mainstream view is that there is a theory of everything that exists but is unknown to us, and that our various theories are approximations of that theory under different assumptions.
Oh! You're so close to seeing the point... There are multiple levels of approximation (at least two), and the one we all experience is Newtonian. Perhaps more accurately, our senses mostly believe pre-Newtonion approximations, which is why it took until Newton to realize how inaccurate they were.
> Also, I don’t see how the logic is fundamentally different between e.g. quantum mechanics and general relativity.
You're pretty radically moving the goalposts here. GP was talking about Newtonian mechanics, not Hamiltonian.
The Hamiltoninan (and Lagrangian) are much more amenable to actual physical calculations, at least on a computer, than the Newtonian formulations of classical mechanics - but otherwise they are perfectly equivalent mathematically. I'm not sure where you'd need any kind of dynamical laws in the building of a cabinet, on the other hand. Are you trying to arrange for a system of inclined planes and pullies to slot the pieces into place?
> Perhaps more accurately, our senses mostly believe pre-Newtonion approximations, which is why it took until Newton to realize how inaccurate they were.
This is a bit of misnomer. Our senses and intuitions are in fact remarkably accurate for a certain range of values, and quite equivalent to what Newton's laws of motion say about these. To some extent, Newton "only" found a simple formalism to represent our existing intuitions. Our intuitions of course break down in other places, such as at very high speeds, or very high altitudes , where relativistic corrections start to become significant.
QM however is a paradigm shift in how the world is described, and it is completely non-intuitive, even in regimes where its predictions are fully aligned with our intuitions and senses. You can use QM to compute the collision of two ideal balls on an ideal plane, and the results will exactly match your intuitions. But the computation and even the representation of the system will not, in any way.
We do that every day using things like finite elements. It’s just a different Hamiltonian that accounts for the fact that we simplify bunches of atoms into a continuum.
> Again, 99.999% of all functional mechanics don't involve epistemology.
Discussing the nature of scientific theories is epistemology. The parent’s point is epistemological in nature.
> ! You're so close to seeing the point... There are multiple levels of approximation (at least two), and the one we all experience is Newtonian.
You are off base. There are many, many approximations that may or may not overlap. It’s not just onion layers.
> You're pretty radically moving the goalposts here. GP was talking about Newtonian mechanics, not Hamiltonian.
GP was talking about different theories applying at different scales. Sorry, it may not have been clear in context but Hamiltonian here is not just the generalisation of Newtonian mechanics we learn about in 2nd year Physics. these theories (quantum mechanics, Newtonian mechanics, and relativity) can be written using the Hamiltonian formalism.
The one we all experience is Aristotelian.
As the past recedes, "the golden age" advances in time. "Hollaback Girl" is now a classic oldie.
So “ Baroque” is J.S. Bach, Handel, Charpentier, Purcell etc (1600-1750 ish), “classical” is mozart, J.C. Bach, Hayden etc up to Beethoven (1750ish-1800ish).
Classical mechanics as I was taught it is not to do with the time period,it means “mechanics when you don’t need to worry about relativity or quantum anything “.
I mean Newton wouldn’t recognize most of what we teach as Calculus because we use Leibnitz’ notation[1] or Lagrange’s notation almost exclusively, and only use Newton’s notation for derivatives with respect to time and don’t ever use Newton’s terminology (fluxions etc).
[1] So named because it was mostly invented by Euler.
Failing a full derivation from the ground up, a proof of the equivalence to Newtonian mechanics would be interesting.
But if your question was more about "why minimizing something yields trajectories", I personally would argue this is beyond physics. As an empirical science, physicists have seen this kind of behaviour broadly (optics, classical mechanics, quantum mechanics) and just unified it as an overarching principle.
Finally regarding the proof to newtonian mechanics, I don't have anything handy from the pure Newtonian perspective beyond the usual "minimises the lagrangian and your equations of motions look the same". However, you might be interested in proofs which show newtonian gravity as low energy approximation of general relativity. And since general relativity has a nice action formulation, it all gets nicely tied in.
Hope this helps!
My motivation for getting to the bottom of all this is to fill the gaps in my physics understanding at least up to quantum mechanics. I have a grasp of QM but I would like to have some insight into the conceptual leaps that brought us there from classical mechanics. QM works in the Hamiltonian picture and I recall from my undergrad days that you get there from a Legendre transformation on the Lagrangian (or something to that effect) so I'm trying to understand the justification of that approach before moving up the conceptual ladder.
Ideally I would like to be able to trace my way from simple postulates based on observation of the physical world all the way to QM, then maybe to QFT after that.
Physics Explained Ultraviolet Catastrophe: https://youtu.be/rCfPQLVzus4
This Veritassium Video goes back further in time and talks about Action, and the Genesis of this idea. https://youtu.be/qJZ1Ez28C-A
Chemistorian, The history of Atomic Theory. https://youtu.be/SqYPrA7upiE
That is only one approach to QM. As you rightly point out, Hamiltonian and Lagrangian approaches are always two sides of the same coin: one is only the Legendre transformation of the other and so they describe the same physics.
So to that end there is a neat QM Lagrangian representation: the path integral formulation. You can apply it to basic QM as well as to QFT or even QFT in curved spacetimes and string theory.
So if your goal is to "trace your way from simple postulates", that is a good way: assume your system can be described by an action, and go from there. It works in pretty much every scenario. In most research I've been involved, you always end up constructing generic actions whose coefficients eventually determine the behaviour of the theory.
And to get back to what I assume is your real conundrum (why do we extremize something to begin with), I just don't think there's any true answer as to why nature behaves this way.
What we can answer is: what is the action, what does it represent, what does it mean to extremize it? The short answer to this (provided by the path integral formulation I mentioned earlier) is that the action is essentially controlling a probability distribution of paths in a given geometry. In quantum mechanics, we interpret this distribution with that of actual particles. When you extremize the distribution, you essentially find the most likely trajectory, and if your distribution is peaked enough around that trajectory, then you can take this path as representative of your system when fluctuations are ignored.
So in the classical limit of QM, that trajectory is all that's left (and that would be the classical mechanics trajectory).
Interestingly a similar interpretation exists in statistical physics. If you "complexify" your time dimension, your action is again on a Euclidean (instead of lorentzian) spacetime and the time direction behaves like a circle whose radius sets a scale akin to a temperature. This might sound a bit complex but where I'm going with this is that once again, you can think of this euclidean path integral as a distribution of paths over fluctuations (this time thermal), and the extremum is this time the system's behaviour when at equilibrium.
Having read about some of the history of this idea, it seems to have been originally built on philosophical grounds, the idea that nature chooses the most harmonious path, as opposed to Newton's laws which seem to come from intuition based on observation of the world. If you keep asking "why?" in either framework you will eventually run up against an epistomological barrier which is unlikely to ever be crossed but in the case of Newton's laws, their basis in physical intuition makes them much easier (for me at least) to accept as given and take as a starting point for constructing a world model. With this being the case I think an acceptable result for me would be to find a proof of equivalence between the Newtonian and Lagrangian pictures. From my reading it seems like the derivation from D'Alambert's principle may be part of the journey.
The modern concept of 'work done' was formulated around 1850 (Eighteen-fifty). That is, we shouldn't assume that back in the days of Lagrange d'Alembert's principle was understood in the same way as it is today.
Joseph Louis Lagrange motivated his notion of potential energy in terms of d'Alembert's principle.
The recurring theme is the concept of 'work done'.
In case you hadn't noticed yet, I'm the contributor who notified you of a resource I created, with interactive diagrams.
There is this distinction: the work-energy theorem expresses physical motion, whereas d'Alembert's virtual work expresses, as the modern name indicates, virtual work.
My assessment is that using d'Alembert's virtual work is an unnecesarily elaborate approach. The same result can be arrived at in a more direct way.
There is a very interesting approach in the quantum physics book by Eisberg and Resnick, section 5.2
To arrive at the Schrödinger equation Eisberg and Resnick construct what they refer to as a plausibility argument.
The goal: to arrive at a wave equation that when solved for the Hydrogen atom will have the electron orbitals as set of solutions.
Eisberg and Resnick state 4 demands:
-1. Must be consistent with the de Broglie/Einstein postulates. wavelength=h/p, frequency=E/h
-2. Must be such that for a quantum entity followed over time the sum of potential energy and kinetic energy is a conserved quantity.
-3. Must be such that the equation is linear in \Psi(x,t): any linear combination of two solutions \Psi_1 and \Psi_2 must also be a solution of the equation. (Motivation: in experiments electron diffraction effects are observed. Interference effects can occur only if wave functions can be _added_.)
-4. In the absence of a potential gradient the equation must have as a solution a propagating sinusoidal wave of constant wavelength and frequency.
Eisberg and Resnick proceed to show that the above 4 demands narrow down the possibilities such that arriving at the Schrödinger equation is made inevitable.
To me the second demand is particularly interesting. The second demand is equivalent to demanding that the work-energy theorem holds good. The recurring theme: the work-energy theorem.
I have a (html)-transcript of the Eisberg & Resnick treatment that I can make available to you.
There is a youtube video with a presentation that is based on the Eisberg & Resnick plausibility argument.
In that video the presentation of the plausibility argument is in the first 18 minutes, the rest of the video is about application of the Schrödinger equation.
There's a PDF here: https://pages.jh.edu/rrynasi1/PhysicalPrinciples/literature/...
[1] An appetising quote:
> The author is well aware that he could have shortened his exposition considerably, had he started directly with the Lagrangian equations of motion and then proceeded to Hamilton’s theory. This procedure would have been justified had the purpose of this book been primarily to familiarize the student with a certain formalism and technique in writing down the differential equations which govern a given dynamical problem, together with certain “recipes” which he might apply in order to solve them. But this is exactly what the author did not want to do. There is a tremendous treasure of philosophical meaning behind the great theories of Euler and Lagrange, and of Hamilton and Jacobi, which is completely smothered in a purely formalistic treatment, although it cannot fail to be a source of the greatest intellectual enjoyment to every mathematically-minded person. To give the student a chance to discover for himself the hidden beauty of these theories was one of the foremost intentions of the author.
The most coherent explanation I've heard was from Feynnman [0]. As far as I understand it (and I may well not have understood it at all well), at the quantum level, all paths are taken by a particle but the contributions of the paths away from the stationary point tend to cancel each other. So, at a macroscopic level, the net effect appears to be be that the particle is following the path of least action.
> a proof of the equivalence to Newtonian mechanics
The Lagrangian method isn't really equivalent to Newton's method. Again, Feynman talks about this in [0]. It's that for a certain class of action, the Euler-Lagrange equations are equivalent to Newton's laws.
It's perfectly plausible to come up with actions that recover systems that represent Einsteinian relativity or quantum mechanics. This is the main reason (as I understand it) why it's considered a more powerful formalism.
The universe has these weird laws, and for now, all we can do is accept them as is. But hopefully, in the future, someone will figure out deeper and simpler principles.
The Hamiltonian approach in contrast is breadth-first: you single out a time axis, start with some initial state, and consider all possibilities that a particle (or field in QFT) could evolve forwards in time just a tiny bit (this is what the Hamiltonian operator does). Then you add up all these possibilities to find the next state, and so you move forwards through time by keeping track of all possible evolutions all at once. This massive superposition of everything that is possible (with corresponding amplitudes) is what you call a state (or wavefunction) and the space that it lives in is the Hilbert (or Fock) space.
So Lagrangian/path-integral: follow full trajectories, then add up all possible choices. depth-first
Hamiltonian/time-evolution: add up all choices for a tiny step in time, then simply do more steps: breadth-first
I imagine it a bit like a scanline algorithm calculating an image as it moves down the screen (Hamiltonian) vs something like a stochastic raytracer that can start with an empty image and refine it pixel by pixel by shooting more rays (Lagrangian)
This is my layman explanation anyways...hopefully it helps, even though i can't say much about their relationship in classical physics.
That is, the relations between the various formulations of classical mechanics are all bi-directional.
At the hub of it al is the work-energy theorem.
I created a resource with interactive diagrams. Move a slider to sweep out variation. The diagram shows how the kinetic energy and the potential energy respond to the variation that is applied.
Starter page: http://cleonis.nl/physics/phys256/stationary_action.php The above page features a case that allows particularly vivid demonstration. An object is launched upwards, subject to a potential that increases with the cube of the height. The initial velocity was tweaked to achieve that after two seconds the object is back to height zero. (The two seconds implementation is for alignment with two other diagrams, in which other potentials have been implemented; linear and quadratic.)
Article with mathematical treatment: http://cleonis.nl/physics/phys256/energy_position_equation.p...
To go from F=ma to Hamilton's stationary action is a two stage proces:
- Derivation of the work-energy theorem from F=ma
- Demonstration that in cases such that the work-energy theorem holds good Hamilton's stationary action holds good also.
General remarks: In the case of Hamilton's stationary action the criterion is: The true trajectory corresponds to a point in variation space such that the derivative of Hamilton's action is zero. The criterion derivative-is-zero is sufficient. Whether the derivative-is-zero point is at a mininum or a maximum of Hamilton's action is of no relevance; it plays no part in the reason why Hamilton's stationary action holds good.
The true trajectory has the property that the rate of change of kinetic energy matches the rate of change of potential energy. Hamilton's stationary action relates to that.
The power of an interactive diagram is that it can present information simultaneously. Move a slider and you see both the kinetic energy and the potential energy change in response. It's like looking at the same thing from multiple angles all at once.
One section of that will be replaced in a day or two: the last part of section 2. I completed a new diagram, that diagram will allow me to cut a lot of text. I believe the change will be a significant improvement.
Repeating the links: Page dedicated to the case of a potential proportional to the cube of the displacement: http://cleonis.nl/physics/phys256/stationary_action.php
From F=ma to Hamilton's stationary action: http://cleonis.nl/physics/phys256/energy_position_equation.p...
There are other demonstrations available that go from the newtonian formulation to Hamilton's stationary action. I believe the one in my resource is the most direct demonstration. (As in: a more direct path doesn't exist, I believe.)
(If you are interested, I can give links to the other demonstrations that I know about.)
https://stackoverflow.com/questions/62518079/scmutils-for-si...
Easter egg: if you dig deeper into the code you will find the "amazing bug" (footnote on page 5)
(Sorry, it's been a while, but iirc the code comments call it the amazing bug, with credit to Radul)
https://github.com/kkylin/mit-scheme-intel-mac-patch?tab=rea...
Works well on Intel Macs and (with effort) mostly works on Apple Silicon.
I was able to make "CloJupyter" notebooks with the examples from the book. You can see some of my notes here (only goes till Chapter 2): https://www.thomasantony.com/projects/sicm-workbook/