Structure and Interpretation of Classical Mechanics (2015)(mitpress.mit.edu)
mitpress.mit.edu
Structure and Interpretation of Classical Mechanics (2015)
https://mitpress.mit.edu/sites/default/files/titles/content/sicm_edition_2/toc.html
91 comments
The description in this article is great, but the why is still rather mysterious. How would somebody come up with that?
If you are familiar with the method of Lagrange multipliers, then what's happening can be explained as follows. Given the Lagrangian L(x,v) the problem of classical mechanics is to find a trajectory x(t),v(t) that extremises the integral of L(x(t),v(t))dt under the constraint x'(t) = v(t). Lagrange multipliers are a method to deal with constraints in optimisation problems. Usually it's taught in the finite dimensional case, but it also works in the infinite dimensional case. We introduce a Lagrange multiplier p(t) and add the constraint to the objective: integral of L(x(t),v(t)) + p(t)(x'(t) - v(t)) dt. To solve the problem we minimise this over x,v,p. If we carry out the minimisation over v first then we're left with two variables x,p. That's the Hamiltonian formulation of the problem, and it's called the dual problem in convex optimisation. So the momentum p is the Lagrange multiplier for the constraint x' = v.
In more detail: we rewrite L(x(t),v(t)) + p(t)(x'(t) - v(t)) = L(x(t),v(t)) - p(t)v(t) + p(t)x'(t). Now we separate out H(x,p) = min_v L(x,v) - pv, so the original problem becomes to minimise the integral of H(x,p) + p(t)x'(t). After applying the Euler-Lagrange equations we obtain Hamilton's equations:
dH/dx = dp/dt
dH/dp = -dx/dt
If you are familiar with the method of Lagrange multipliers, then what's happening can be explained as follows. Given the Lagrangian L(x,v) the problem of classical mechanics is to find a trajectory x(t),v(t) that extremises the integral of L(x(t),v(t))dt under the constraint x'(t) = v(t). Lagrange multipliers are a method to deal with constraints in optimisation problems. Usually it's taught in the finite dimensional case, but it also works in the infinite dimensional case. We introduce a Lagrange multiplier p(t) and add the constraint to the objective: integral of L(x(t),v(t)) + p(t)(x'(t) - v(t)) dt. To solve the problem we minimise this over x,v,p. If we carry out the minimisation over v first then we're left with two variables x,p. That's the Hamiltonian formulation of the problem, and it's called the dual problem in convex optimisation. So the momentum p is the Lagrange multiplier for the constraint x' = v.
In more detail: we rewrite L(x(t),v(t)) + p(t)(x'(t) - v(t)) = L(x(t),v(t)) - p(t)v(t) + p(t)x'(t). Now we separate out H(x,p) = min_v L(x,v) - pv, so the original problem becomes to minimise the integral of H(x,p) + p(t)x'(t). After applying the Euler-Lagrange equations we obtain Hamilton's equations:
dH/dx = dp/dt
dH/dp = -dx/dt
Agreed, thanks. That blog post was just trying to explain what it is, not why it is, mostly as a basis to complain about education rather than teach physics.
The why is actually quite simple- and the same reason the Fourier transform is sometimes used. Some problems are simply more elegantly expressed in a particular basis system. Nobody tries to express a ball's motion in flight via Fourier analysis, but you certainly could.
In the same way, sometimes solutions for the position(s) of a system is the most natural basis system for describing/investigating a problem (use the Lagrangian) and sometimes solutions for its momentum are (use the Hamiltonian).
Of course, they're intrinsically linked since the evolution of one determines the evolution of the other.
In the same way, sometimes solutions for the position(s) of a system is the most natural basis system for describing/investigating a problem (use the Lagrangian) and sometimes solutions for its momentum are (use the Hamiltonian).
Of course, they're intrinsically linked since the evolution of one determines the evolution of the other.
Christ, that's _awesome_. I wish I was taught the Legendre transform this way. Where did you run into this viewpoint? I'd love to read a textbook that explains physics this way.
In fact, I to this day have not found a book that caters to my level of mathematical knowledge (undergrad pure math) on variational calculus. They're either _way_ too handwavy (like Taylor), or _way_ beyond reach (relying on heavy functional analysis, weak convergence, all that stuff).
Is there a textbook/lecture series/what have you that explains the calculus of variations at a "non-handwavy" level, BUT does not tries to perform the calculations in the most general way possible. That is, pick a nice space (C^\infinity or some such) and then show me with full rigor how the calculus of variations can be constructed. I'd love some references :)
In fact, I to this day have not found a book that caters to my level of mathematical knowledge (undergrad pure math) on variational calculus. They're either _way_ too handwavy (like Taylor), or _way_ beyond reach (relying on heavy functional analysis, weak convergence, all that stuff).
Is there a textbook/lecture series/what have you that explains the calculus of variations at a "non-handwavy" level, BUT does not tries to perform the calculations in the most general way possible. That is, pick a nice space (C^\infinity or some such) and then show me with full rigor how the calculus of variations can be constructed. I'd love some references :)
I don't think I've seen this viewpoint anywhere else, but Langrange multipliers in the calculus of variations are a well known thing, and the connection between Lagrange multipliers and the Legendre transform is a well known thing, so I'm sure you can find it in a book somewhere.
I've not read any book on the calculus of variations, but I think that the book by Gelfand and Fomin is the standard book. Maybe that is suitable?
By the way, you can motivate the Legendre transform / Langrange multipliers like this. Suppose we want to minimise f(x) subject to g(x) = 0. We can add the constraint g(x) = 0 to the objective function by defining Q(x) = 0 if g(x) = 0 and Q(x) = infinite otherwise. Then the problem is min f(x) + Q(x). This seems like the kind of thing that doesn't help, but now write Q(x) = max_p (p g(x)). You can see that this is indeed the same as the previous definition of Q. So we can write that problem as min_x max_p f(x) + p g(x). This is a bit silly, and the usual way to explain Lagrange multipliers is probably better, but you may find it amusing.
I've not read any book on the calculus of variations, but I think that the book by Gelfand and Fomin is the standard book. Maybe that is suitable?
By the way, you can motivate the Legendre transform / Langrange multipliers like this. Suppose we want to minimise f(x) subject to g(x) = 0. We can add the constraint g(x) = 0 to the objective function by defining Q(x) = 0 if g(x) = 0 and Q(x) = infinite otherwise. Then the problem is min f(x) + Q(x). This seems like the kind of thing that doesn't help, but now write Q(x) = max_p (p g(x)). You can see that this is indeed the same as the previous definition of Q. So we can write that problem as min_x max_p f(x) + p g(x). This is a bit silly, and the usual way to explain Lagrange multipliers is probably better, but you may find it amusing.
[deleted]
As someone who struggled dealing with physicists explanations in college I argue that much of the lack of clarity is a deliberate ambiguity as to the meaning of the underlying mathematical structure of the symbols you're informed to manipulate.
As an example, when I was taught calculus of variations, we didn't dwell too much on the fact the symbols in our functions we were differentiating were no longer corresponding to points in R^n, but rather to functions existing in function space. Why the mathematics we were taught in Vector Calculus worked here as well was quietly ignored, to the extent that when I raised the issue with classmates, they didn't realize that anything had changed from vector calculus.
As another example, is physicists playing fast and loose with what exactly they mean by 'vector' (hint: they mean it's really a function on R^3, but good luck finding an undergrad textbook that says that; I'm curious as to how Sussman treats this).
Sussman, by virtue of implementing programs, takes a constructive approach, and thus forces both the author and reader to come to an explicit understanding of what the underlying mathematical objects actually are and the structure of how solutions are formed. Thus, I am not surprised that he comes up with a superior and clearer definition of the Legendre transform.
As an example, when I was taught calculus of variations, we didn't dwell too much on the fact the symbols in our functions we were differentiating were no longer corresponding to points in R^n, but rather to functions existing in function space. Why the mathematics we were taught in Vector Calculus worked here as well was quietly ignored, to the extent that when I raised the issue with classmates, they didn't realize that anything had changed from vector calculus.
As another example, is physicists playing fast and loose with what exactly they mean by 'vector' (hint: they mean it's really a function on R^3, but good luck finding an undergrad textbook that says that; I'm curious as to how Sussman treats this).
Sussman, by virtue of implementing programs, takes a constructive approach, and thus forces both the author and reader to come to an explicit understanding of what the underlying mathematical objects actually are and the structure of how solutions are formed. Thus, I am not surprised that he comes up with a superior and clearer definition of the Legendre transform.
Having tought classical mechanics I have to say, yes we do that (intentionally), but it is due to a lack of time and due to a lack of preparedness of the students. I have 16 weeks of instruction, with 2 lectures of 90 minutes each, to cover all the parts of classical mechanics that the students will need later. (Very little research is done in classical mechanics these days). A third of the class is deadly afraid of calculus. I simply can not spare 6 lectures to explain caculus of variatians, when I only need it for two derivations. After all it still is a a vector space and the math they know still works. I am happy to discuss the details (why does it work, what is a function space, what is the dual space of distributions) with the five students who care after class or during office hours.
When teachers do this, they teach students to think that having a fuzzy understanding about the foundations of the subject is normal rather than something that should raise an internal alarm bell. At the very least, teachers should minimize damage by announcing clearly when they've stopped explaining and started just giving a prescription, and point the students to the place where they can actually learn what's going on.
The same issue comes up on teaching probability, and even worse, elementary stochastic processes. To do it correctly, you need measure theory, but that requires a whole course or courses.
In fact, a lot of the results of freshman calculus require more mathematical analysis than most students really have command of. But we still teach it because the applications are so compelling, and the intuition is important to develop.
Even more generally, as you probably know, there are tricky foundational issues in Mathematics that most professional mathematicians prefer not to pay attention to.
In fact, a lot of the results of freshman calculus require more mathematical analysis than most students really have command of. But we still teach it because the applications are so compelling, and the intuition is important to develop.
Even more generally, as you probably know, there are tricky foundational issues in Mathematics that most professional mathematicians prefer not to pay attention to.
Well I did that. I also gave them a similar warning when I started using the dirac delta, because it is not a proper function and if you want to know more than "it eats integrals", you have to either view it as the limit in a series of function or as a distribution acting on test functions. The thing is you walk a fine line between hiding the truth and not cranking the internal alarm of the students up to a volume where is scares them and prevents further learning.
That last sentence sounds like a euphemism for there simply being a conflict with what students want to learn about and what instructors are required to teach. How would I tell the difference between students being scared versus them healthily resisting the application of mechanical procedures they don't understand?
In the first case you would get outright rejection (typically expressed as "why do we even need that" or "nobody can understand that"), in the latter at least the better students would ask question "why can we do that step", "why does this follow", "earlier you said X, not we do Y, why?".
OK, you just mean that different students will want different levels of rigor/clarity. Sure.
Yes. And better students tend to want more rigor (or have more brain cycles left over for more rigor in the introduction of new but only tangentially relevant topics) than the weaker students, who are usually more happy to take something at face value as long at it let's them continue with the main topic. The line you have to walk as a teacher is then to find an appropriate level of rigor to not loss (too many) students, while satisfying the curriosity of the more interested students and telling them where to find the extra details they might want to know.
OK but I would describe that as the students having already been poorly partitioned into classes. If you're taking a class where you need to go read independently to understand what's going, it seems much better to either just read fully independently or to take a class in the actual thing you want to learn. Sending a bunch of kids with widely varying levels of skill and interest through the same course is hugely wasteful, however common that may be.
This is not necessarily a wrong thing to teach. As one progresses from old things that are well-understood to new developments,students will have to deal with fuzziness and gaps. They must learn to make progress using a heuristic feel for the subject instead of complete rigour. Too sensitive an alarm for missing foundations will actually stop them from being productive. Think of how much time passed between the invention of calculus and the development of rigorous analysis.
That's probably appropriate for quickly progressing fields, but not for fundamental physics which has utterly stagnated.
People wasting time on important but nearly intractable topics certainly happens (e.g., arguably myself on quantum foundations). But this is absolutely dwarfed by the number of researchers eager to springboard to a cutting edge, which hasn't moved much in decades, but who remain ignorant of the basics.
The terrible incentives for professors to quickly make graduate students useful, rather than invest in fundamental understanding that won't pay off for many years after they graduate, should also lead us to suspect the correct balance isn't being struck.
The terrible incentives for professors to quickly make graduate students useful, rather than invest in fundamental understanding that won't pay off for many years after they graduate, should also lead us to suspect the correct balance isn't being struck.
Reminds me: where I smashed my head against the wall with mechanics wasn't the calculus part, but when I encountered L = T - U and asked "Why?". I was referred to the book, which in turn helpfully explained "You are certainly entitled to ask why the quantity T - U should be of any interest. There seems to be no simple answer to this question except that it is."
Yea, I agree with your diagnosis. I even mention the calculus of variations as an example (and give a book recommendation) in a footnote in the post.
[deleted]
I don't know how many times I read over the definition of the Legendre transform and never really got it. I'll definitely check out that section. I feel like as an outsider Sussman brings a much needed fresh viewpoint in explaining things in physics. His explanation of the derivatives in the Euler-Lagrange equation was similarly great.
I don't understand this mysticism. Legendre transformation is just one example of canonical transformations that you can do switch from one set of variables to another, typically out of convenience (you can rest assured physicists are aware of this, and I would be more conservative before saying it lies "at the heart" of anything --it just links Hamiltonian mechanics to Lagrangian mechanics, but they're basically two different ways we use to describe the same physical thing, each parametrization with their own intuitive features--, and it doesn't really show up in quantum mechanics).
Yes, you can go beyond the actual purpose of "I just want to get a function of x,y starting from a function of x,z" and do illustrative geometric interpretations for every single canonical transformation out there, but physics books tend be practical and not to dwell on such mathematical "curiosities".
Still, Legendre transform is arguably the most important one, and some books do go further. On top of my mind, you can see Analytical Mechanics by Hand & Finch, preceding that blog post by 2 decades. (Interestingly, he claims to have read this book --not carefully, apparently, because the equation he finds surprising is there; it's just written in words relating two equations, rather than as a new symbolic equation).
Yes, you can go beyond the actual purpose of "I just want to get a function of x,y starting from a function of x,z" and do illustrative geometric interpretations for every single canonical transformation out there, but physics books tend be practical and not to dwell on such mathematical "curiosities".
Still, Legendre transform is arguably the most important one, and some books do go further. On top of my mind, you can see Analytical Mechanics by Hand & Finch, preceding that blog post by 2 decades. (Interestingly, he claims to have read this book --not carefully, apparently, because the equation he finds surprising is there; it's just written in words relating two equations, rather than as a new symbolic equation).
You mistakenly say the Legendre Transform doesn't really show up in quantum mechanics, which is likely because you didn't notice when it appeared in the construction of the path integral. (Or maybe you just haven't used quantum mechanics outside of the non-relativistic domain, in which case you've probably only used the Hamiltonian formulation and don't know why? I don't know your background.) That's an easy mistake to make if you think about it as just some coordinate transform, since then you won't notice it when it's used in a different guise than when it was taught to you. Many field theorists would say the path integral is the most beautiful idea in theoretical physics -- Noether's theorem probably being the only idea that is more popular -- and if you don't understand how it bakes in the Legendre transform you don't understand it well.
Other than that technical mistake, which I think is revealing, the rest of your comment is hard to respond to because it takes the form "this thing you claim is important for given reasons just isn't; it's not a big deal". So all I can do is try to flesh out my reasons, and hope you'll engage with them.
The Lagrangian and Hamiltonian formulations are the two primary ways to write down fundamental physical laws. There is no third formulation of remotely comparable importance or clarity, so the Legendre transform has a special role not enjoyed by other coordinate transforms. If you want to actually understand why laws are formulated how they are, and especially if you want to be prepared to move beyond them in case new physics requires it, you need to have a deep understanding of how the Lagrangian and Hamiltonian formulations are linked. Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?" "Where does the incredible richness of symplectic geometry come from, and why isn't there anything similar associated with the Lagrangian formulation?" "How would any of this change if there was more than one time dimension?"
Regardless, Hand & Finch is awful. I encourage you to quote the piece of Hand & Finch on the Legendre Transform that you think is clear, as I have quoted the books I think are confused. And then we can see whether students find my explanation or their explanation clearer.
If you say "well, the ideas are encoded in Hand & Finch even if the students don't understand them", you've missed the point. I make zero claims of novelty, and neither, I'm sure, would Hand & Finch. (These ideas are nearly two centuries old, so pointing out that their textbook is two decades old is silly.) My complaint, rather, is that these old ideas crucial to the invention of mechanics are not being faithfully transmitted to generations of students who take mechanics for granted.
Other than that technical mistake, which I think is revealing, the rest of your comment is hard to respond to because it takes the form "this thing you claim is important for given reasons just isn't; it's not a big deal". So all I can do is try to flesh out my reasons, and hope you'll engage with them.
The Lagrangian and Hamiltonian formulations are the two primary ways to write down fundamental physical laws. There is no third formulation of remotely comparable importance or clarity, so the Legendre transform has a special role not enjoyed by other coordinate transforms. If you want to actually understand why laws are formulated how they are, and especially if you want to be prepared to move beyond them in case new physics requires it, you need to have a deep understanding of how the Lagrangian and Hamiltonian formulations are linked. Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?" "Where does the incredible richness of symplectic geometry come from, and why isn't there anything similar associated with the Lagrangian formulation?" "How would any of this change if there was more than one time dimension?"
Regardless, Hand & Finch is awful. I encourage you to quote the piece of Hand & Finch on the Legendre Transform that you think is clear, as I have quoted the books I think are confused. And then we can see whether students find my explanation or their explanation clearer.
If you say "well, the ideas are encoded in Hand & Finch even if the students don't understand them", you've missed the point. I make zero claims of novelty, and neither, I'm sure, would Hand & Finch. (These ideas are nearly two centuries old, so pointing out that their textbook is two decades old is silly.) My complaint, rather, is that these old ideas crucial to the invention of mechanics are not being faithfully transmitted to generations of students who take mechanics for granted.
Ugh. So much noise. I was going to avoid this but I'll bite once. I'm a professor in condensed matter physics, doing research actively, quite well aware of Lagrangian mechanics and relativistic QFT, path integrals, and I also taught QFT. That's my background.
You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian". No. And no, saying no isn't a "technical mistake". First, when building theories from symmetries, you typically start from a Lagrangian as the fundamental quantity, as opposed to taking the Legendre transform of something. Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. It's useful before you do a canonical quantization of a classical theory, because you can only do Legendre transform between classical Hamiltonian or Lagrangian, but that's still something you do when doing classical mechanics, not quantum mechanics. Even then you need to be careful with noncommuting operators --Legendre transform won't magically give you the correct Lagrangian and you need to do some trial and error to find the correct form (which happens even with the QED Lagrangian with derivatives and field tensor terms). Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong.
Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well. And no, you don't need path integrals for doing relativistic QM/QFT either.
There are third and fourth and other formulations of mechanics (Louville theorem, Hamilton-Jacobi equation, ...). You're just dismissive about them because they don't serve your narrative.
Hand & Finch reads perfectly fine to me, everything is there, it's not "encoded", you just need to read the equation and the text. There will always be confused students, even if you spell everything out.
And this whole attitude reminds me of Feynman's comment on mathematical physicists or mathematicians doing physics. I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.
You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian". No. And no, saying no isn't a "technical mistake". First, when building theories from symmetries, you typically start from a Lagrangian as the fundamental quantity, as opposed to taking the Legendre transform of something. Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. It's useful before you do a canonical quantization of a classical theory, because you can only do Legendre transform between classical Hamiltonian or Lagrangian, but that's still something you do when doing classical mechanics, not quantum mechanics. Even then you need to be careful with noncommuting operators --Legendre transform won't magically give you the correct Lagrangian and you need to do some trial and error to find the correct form (which happens even with the QED Lagrangian with derivatives and field tensor terms). Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong.
Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well. And no, you don't need path integrals for doing relativistic QM/QFT either.
There are third and fourth and other formulations of mechanics (Louville theorem, Hamilton-Jacobi equation, ...). You're just dismissive about them because they don't serve your narrative.
Hand & Finch reads perfectly fine to me, everything is there, it's not "encoded", you just need to read the equation and the text. There will always be confused students, even if you spell everything out.
And this whole attitude reminds me of Feynman's comment on mathematical physicists or mathematicians doing physics. I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.
> I'm a professor in condensed matter physics, doing research actively, quite well aware of Lagrangian mechanics and relativistic QFT, path integrals, and I also taught QFT. That's my background.
Sorry, that was my mistake. I didn't expect that someone knowledgeable would say the Legendre transform "doesn't really show up in quantum mechanics". Even if you want to dispute whether the switch between Lagrangian and Hamiltonian formulations of quantum mechanics is truly a Legendre transform, rather than a generalization that doesn't deserve that name, it seems incumbent on you to state clearly that this was your (semantical) argument rather than pretending that this was obvious.
I don't know how a better way to "objectively" establish this than just to Google "path integral legendre transform"; the first hit is arXiv:1612.00462 (in a respectable journal by respectable authors, etc.) whose first sentence is "Legendre transforms play important roles in quantum field theory" before going on to review their central role in the divergence issues that appear immediately when you introduce perturbative expansions in QFT. Neither of us care about this random paper, but it just doesn't seem reasonable to dismiss Legendre transforms as not really showing up without further qualification.
Anyways, that was just to explain my mistaken assumption about your background. Nonetheless, I do apologize as I should have been more careful. I didn't mean to cause offense.
> You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian"
No, I'm saying that if you have the same quantum system (including field theories) that you can treat in the Lagrangian and Hamiltonian formulations, the connection between them makes use of the Legendre transform.
> And no, saying no isn't a "technical mistake".
Now that I understand better what you were trying to say, I withdraw that language with apologies. I still think you're mistaken, but it's about aesthetics, semantics, and pedagogy rather technical assertions.
> First, when building theories from symmetries, you typically start from a Lagrangian as the fundamental quantity, as opposed to taking the Legendre transform of something.
It's certainly true that we can build field-theoretic Lagrangians from symmetries and calculate important quantities without touching a Legendre transform. But the same can be said about classical mechanics, and it doesn't mean the Legendre transform doesn't have a central organizing role in our body of knowledge.
> Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. ... Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong.
I definitely did not suggest that one could do a Legendre transform of operators as if they were commuting variables. But saying that the Legendre transform doesn't play a role in quantum systems for that reason is like saying Maxwell's equations don't play a role in QED because E and B are commuting variables in Maxwell's equations.
> Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well.
Yes, I'm well aware. The reason I parenthetically asked "Or maybe you just haven't used quantum mechanics outside of the non-relativistic domain...?" is because most students first use it in a relativistic QFT course (rather than just seeing it in Sakurai). I should have said "field-theoretic context" to be clearer, although of course you can use them even without fields.
> And no, you don't need path integrals for doing relativistic QM/QFT either.
Sure, you can do some QM/QFT without path integrals just like you can do classical mechanics solely in the Lagrangian framework or solely in the Hamiltonian framework. But, unlike classical mechanics and unlike quantum field theories where there is a larger diversity of techniques, non-relativistic QM with a finite number of degrees of freedom is usually done solely in the Hamiltonian framework, so people rarely think about the connection to Lagrangians.
> There are third and fourth and other formulations of mechanics (Louville theorem, Hamilton-Jacobi equation, ...). You're just dismissive about them because they don't serve your narrative.
Louville's theorem is a result within Hamiltonian mechanics. I've never heard anyone suggest that it's a distinct formulation/framework, and I'm honestly confused as to what you could mean by this. Can you explain your viewpoint more, or point me toward somewhere where I could read more about it?
(Or do you just mean using Louville's equation to evolve a probability distributions over phase space rather than Hamilton's equation to evolve individual phase-space points? This is no less Hamiltonian mechanics than probabilistic Turing machines are Turing machines.)
Whether the Hamilton-Jacobi equation qualifies as a similarly fundamental formulation as the Lagrangian or Hamiltonian formulations is at least an arguable assertion, although one I certainly disagree with. (Recall my "remotely similar importance" criteria.) I'd be happy to debate you on this, but it sounds like you're mostly uninterested in aestetic/ontological questions, e.g., what makes a formulation qualify as "fundamental". That's fine, you don't have to care about those debates to do good science, but that's the nature of our disagreement here, and I think you would need to address it before saying I'm just trying to serve my narrative. (Honestly, what ulterior purpose do you think I have? Scoring fake internet point by writing blog post rants?)
> Hand & Finch reads perfectly fine to me, everything is there,
I previously suggested that you quote the parts you think are clear, as I have quoted the parts I think are confused in other books. (Since you're the one who thinks it's explained well in Hand & Finch, it makes a lot more sense for you to quote the relevant section where you think that happens than for me to quote all the sections where I think it doesn't.) You can certainly dismiss this exercise as not worth your time, but I can't really engage with you on this because you're just asserting the same thing as your first comment, not actually gathering evidence for it.
"Reads perfectly fine to me", a professor, is not a good criterion for deciding which books do a good job of explaining to students. Curse of knowledge and all that.
> it's not "encoded", you just need to read the equation and the text. There will always be confused students, even if you spell everything out.
I'm not saying treatments like Hand & Finch are bad because some confused students exist, I'm saying they're bad because most students remain confused and better, much-less-confusing treatments are possible. I challenged you to compare my explanation to Hand & Finch's by seeing which is actually understood more by students.
To be concrete, I predict that less than 20% of any class of undergraduate students (e.g., Princeton, where I took Hand & Finch with confused classmates) will be able to correctly report that two functions are each other's Legendre transforms when their derivatives are inverses. (If you think that sort of failure rate is just par for the course, then I think we have very different teaching standards and there's not much more to discuss.)
Before I wrote the blog post, I surveyed two active research faculty and three postdocs and none knew this fact. I similarly challenge you to ask your colleagues to explain the Legendre transform on the spot and see the fraction of them that can say something more rigorous than H = pv - L.
> I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.
If you haven't spent any time thinking about it, wouldn't it be better to just not have an opinion one way or the other? "The Legendre transform isn't deep" is a very different assertion to "it's pointless to worry which parts of physics are the deepest".
Regardless, I think you're making a mistake that ultimately will have a small but non-trivial negative impact on your students.
Sorry, that was my mistake. I didn't expect that someone knowledgeable would say the Legendre transform "doesn't really show up in quantum mechanics". Even if you want to dispute whether the switch between Lagrangian and Hamiltonian formulations of quantum mechanics is truly a Legendre transform, rather than a generalization that doesn't deserve that name, it seems incumbent on you to state clearly that this was your (semantical) argument rather than pretending that this was obvious.
I don't know how a better way to "objectively" establish this than just to Google "path integral legendre transform"; the first hit is arXiv:1612.00462 (in a respectable journal by respectable authors, etc.) whose first sentence is "Legendre transforms play important roles in quantum field theory" before going on to review their central role in the divergence issues that appear immediately when you introduce perturbative expansions in QFT. Neither of us care about this random paper, but it just doesn't seem reasonable to dismiss Legendre transforms as not really showing up without further qualification.
Anyways, that was just to explain my mistaken assumption about your background. Nonetheless, I do apologize as I should have been more careful. I didn't mean to cause offense.
> You're mistakenly equating Legendre transform to Lagrangian, and basically saying "there is Legendre transform in quantum mechanics because there's Lagrangian"
No, I'm saying that if you have the same quantum system (including field theories) that you can treat in the Lagrangian and Hamiltonian formulations, the connection between them makes use of the Legendre transform.
> And no, saying no isn't a "technical mistake".
Now that I understand better what you were trying to say, I withdraw that language with apologies. I still think you're mistaken, but it's about aesthetics, semantics, and pedagogy rather technical assertions.
> First, when building theories from symmetries, you typically start from a Lagrangian as the fundamental quantity, as opposed to taking the Legendre transform of something.
It's certainly true that we can build field-theoretic Lagrangians from symmetries and calculate important quantities without touching a Legendre transform. But the same can be said about classical mechanics, and it doesn't mean the Legendre transform doesn't have a central organizing role in our body of knowledge.
> Second, when strictly doing quantum mechanics, Legendre transform itself doesn't make any sense because everything is an operator. ... Thinking that ordinary Legendre transform will just work in quantum mechanics is naive and plain wrong.
I definitely did not suggest that one could do a Legendre transform of operators as if they were commuting variables. But saying that the Legendre transform doesn't play a role in quantum systems for that reason is like saying Maxwell's equations don't play a role in QED because E and B are commuting variables in Maxwell's equations.
> Path integrals don't have anything to do with relativity, they work perfectly fine for non-relativistic QM as well.
Yes, I'm well aware. The reason I parenthetically asked "Or maybe you just haven't used quantum mechanics outside of the non-relativistic domain...?" is because most students first use it in a relativistic QFT course (rather than just seeing it in Sakurai). I should have said "field-theoretic context" to be clearer, although of course you can use them even without fields.
> And no, you don't need path integrals for doing relativistic QM/QFT either.
Sure, you can do some QM/QFT without path integrals just like you can do classical mechanics solely in the Lagrangian framework or solely in the Hamiltonian framework. But, unlike classical mechanics and unlike quantum field theories where there is a larger diversity of techniques, non-relativistic QM with a finite number of degrees of freedom is usually done solely in the Hamiltonian framework, so people rarely think about the connection to Lagrangians.
> There are third and fourth and other formulations of mechanics (Louville theorem, Hamilton-Jacobi equation, ...). You're just dismissive about them because they don't serve your narrative.
Louville's theorem is a result within Hamiltonian mechanics. I've never heard anyone suggest that it's a distinct formulation/framework, and I'm honestly confused as to what you could mean by this. Can you explain your viewpoint more, or point me toward somewhere where I could read more about it?
(Or do you just mean using Louville's equation to evolve a probability distributions over phase space rather than Hamilton's equation to evolve individual phase-space points? This is no less Hamiltonian mechanics than probabilistic Turing machines are Turing machines.)
Whether the Hamilton-Jacobi equation qualifies as a similarly fundamental formulation as the Lagrangian or Hamiltonian formulations is at least an arguable assertion, although one I certainly disagree with. (Recall my "remotely similar importance" criteria.) I'd be happy to debate you on this, but it sounds like you're mostly uninterested in aestetic/ontological questions, e.g., what makes a formulation qualify as "fundamental". That's fine, you don't have to care about those debates to do good science, but that's the nature of our disagreement here, and I think you would need to address it before saying I'm just trying to serve my narrative. (Honestly, what ulterior purpose do you think I have? Scoring fake internet point by writing blog post rants?)
> Hand & Finch reads perfectly fine to me, everything is there,
I previously suggested that you quote the parts you think are clear, as I have quoted the parts I think are confused in other books. (Since you're the one who thinks it's explained well in Hand & Finch, it makes a lot more sense for you to quote the relevant section where you think that happens than for me to quote all the sections where I think it doesn't.) You can certainly dismiss this exercise as not worth your time, but I can't really engage with you on this because you're just asserting the same thing as your first comment, not actually gathering evidence for it.
"Reads perfectly fine to me", a professor, is not a good criterion for deciding which books do a good job of explaining to students. Curse of knowledge and all that.
> it's not "encoded", you just need to read the equation and the text. There will always be confused students, even if you spell everything out.
I'm not saying treatments like Hand & Finch are bad because some confused students exist, I'm saying they're bad because most students remain confused and better, much-less-confusing treatments are possible. I challenged you to compare my explanation to Hand & Finch's by seeing which is actually understood more by students.
To be concrete, I predict that less than 20% of any class of undergraduate students (e.g., Princeton, where I took Hand & Finch with confused classmates) will be able to correctly report that two functions are each other's Legendre transforms when their derivatives are inverses. (If you think that sort of failure rate is just par for the course, then I think we have very different teaching standards and there's not much more to discuss.)
Before I wrote the blog post, I surveyed two active research faculty and three postdocs and none knew this fact. I similarly challenge you to ask your colleagues to explain the Legendre transform on the spot and see the fraction of them that can say something more rigorous than H = pv - L.
> I personally have better things to do than dwelling on mysteries of Legendre transform, or deep meanings and magical connections of the number pi.
If you haven't spent any time thinking about it, wouldn't it be better to just not have an opinion one way or the other? "The Legendre transform isn't deep" is a very different assertion to "it's pointless to worry which parts of physics are the deepest".
Regardless, I think you're making a mistake that ultimately will have a small but non-trivial negative impact on your students.
> Otherwise it's impossible to answer questions like: "If the Lagrangian formulation put space and time on equal footing, and if the Hamiltonian formulation gives a preferred role to time (generating time evolution), could we give a similarly preferred role to space?" "More generally, why isn't there a third or forth major formulation of mechanics?"
Hey, can you answer those questions? Or point to the right answers. Thanks!
Hey, can you answer those questions? Or point to the right answers. Thanks!
I'm mostly unable. It was musing about these questions that got me interested in really understanding the Legendre transform.
I can say that dynamical equations that try to generate spatial translation from initial data on a time-like slice, rather than time translation from a space-like slice like Hamiltonian dynamics, are doomed because there is no well-posed initial-value problem (except in certain special cases involving massless particles), e.g., you generically cannot infer what's far from a spatial plane even if you know everything that happens on that plane for all time. Related topics:
http://www.scholarpedia.org/article/Hyperbolic_dynamics https://en.wikipedia.org/wiki/Well-posed_problem
Also, I would say that the fact that the Legendre transform is a manifest involution gives (quite) weak evidence that there are no other major formulations to find. Of course, it's possible to use a hybrid strategy, Routhian mechanics, with Lagrangian and Hamiltonian formulations on different degrees of freedom:
https://en.wikipedia.org/wiki/Routhian_mechanics
I can say that dynamical equations that try to generate spatial translation from initial data on a time-like slice, rather than time translation from a space-like slice like Hamiltonian dynamics, are doomed because there is no well-posed initial-value problem (except in certain special cases involving massless particles), e.g., you generically cannot infer what's far from a spatial plane even if you know everything that happens on that plane for all time. Related topics:
http://www.scholarpedia.org/article/Hyperbolic_dynamics https://en.wikipedia.org/wiki/Well-posed_problem
Also, I would say that the fact that the Legendre transform is a manifest involution gives (quite) weak evidence that there are no other major formulations to find. Of course, it's possible to use a hybrid strategy, Routhian mechanics, with Lagrangian and Hamiltonian formulations on different degrees of freedom:
https://en.wikipedia.org/wiki/Routhian_mechanics
I don’t think the typical “change of variables” definition is bad.
You take the derivative of L along the fiber of the tangent bundle.
If the derivative is non-singular it defines an isomorphism in each
point of the tangent space with the cotangent space. And that’s the
important thing, going from the tangent bundle to the cotangent
bundle. Now we can use all the beauty of symplectic geometry
The change of variable definition, as actually presented in the textbooks everyone teaches from, is horrible. That's the topic of the blog post. Yes, that definition can be made clear after introducing a bunch of machinery of symplectic geometry, but I'm doubtful this is good pedagogy and I'm confident that, due to time constraints, it could never be taught to most physicists.
Great summary. I also love Prof. V. Balakrishnan's exposition on this topic. The video is long, but he motivates the ideas very clearly and in context: https://www.youtube.com/watch?v=GOkZs2RZMQY.
Your blog is a great resource. Especially the fact that you've linked other blogs and links is really helpful.
As a college sophomore majoring in Physics, I'm curious: what level of student is this text accessible to?
I actually took this class a sophomore in CS and it was part of reason I switched to physics. It's not an easy text - there's a lot you have to keep in your head all at once - but the approach to mechanics is the most elegant I've seen (personal opinion though). It was the first class I took where I really had to read through the book and my notes after class, but the understanding you gain about classic mechanics, math, and functional programming are second to none. Really formative experience I'd say - well worth the effort!
Was this book your first exposure to Classical Mechanics? Would you recommend it as such?
This book (and the class) were my first exposure to Lagrangian and Hamiltonian dynamics and it's served me very well. Background up to that point was high school mechanics and college freshman mechanics and E&M. I think the book alone would be a more challenging first introduction to Lagrangians and Hamiltonians, but it does give a very good "deep" understanding of the material, mostly because you can play around with functions, equations, and systems without having to grind through pages of algebra since the computer does it.
Outside of a class, it might be best approached with a few friends working through it together or in conjunction with one of the more traditional mechanics texts. Definitely worth the read at some point!
Outside of a class, it might be best approached with a few friends working through it together or in conjunction with one of the more traditional mechanics texts. Definitely worth the read at some point!
This is a wonderful article. I'm sure I never really understood what the Legendre transformation was before - not this well.
What other textbooks would you suggest?
As mentioned in a footnote in the blog post, I think Gelfand and Fomin’s “Calculus of Variations” is very clear on its topic. Wald's "General Relativity" is slow-going, but excellent. No good textbooks on quantum mechanics or QFT exist; Weinberg's "Quantum Theory of Fields" is probably the least terrible on that. Nielsen and Chuang's "Quantum Computation and Quantum Information" is still excellent, but pretty out-of-date.
Also, to be clear, I haven't worked through most of Sussman, so I can't recommend it one way or the other. I was just commenting on the handling of the Legendre transform.
Also, to be clear, I haven't worked through most of Sussman, so I can't recommend it one way or the other. I was just commenting on the handling of the Legendre transform.
I would recommend Quantum Field Theory for the Gifted Amateur by Lancaster and Blundell. It has many clear worked examples and builds up lots of the conceptual and technical ideas in a way that felt comfortable. Also the chapters are quite short and are digestible in one sitting.
> No good textbooks on quantum mechanics or QFT exist
No love for Cohen-Tannoudji?
No love for Cohen-Tannoudji?
Haven't read it, but based on the table of contents it looks like it has at least two of the very common problems with QM books: (1) starts with continuous degrees of freedom rather than qubits, and (2) talks about observables rather than PVMs.
This isn't surprising since much of what's wrong with how QM is taught has been illuminated by the young field of quantum information, which hardly existed when the book was published in 1992.
For me talking more about the second point, see here: http://blog.jessriedel.com/2016/11/12/how-to-think-about-qua...
For me talking more about the second point, see here: http://blog.jessriedel.com/2016/11/12/how-to-think-about-qua...
You can play with the software online here: http://io.livecode.ch/learn/namin/scheme-mechanics
The notation chapter with live snippets is reproduced here: http://io.livecode.ch/learn/namin/scheme-mechanics/chapter9
The notation chapter with live snippets is reproduced here: http://io.livecode.ch/learn/namin/scheme-mechanics/chapter9
At first glance, this looks like an excellent textbook.
Personally, my favourite theorem in classical mechanics is the so called 'tennis racket theorem', sometimes known as the 'intermediate axis theorem'.
It explains why objects with roughly three different moments of intertia have unstable rotation about their intermediate moment.
It can be easily demonstrated with a tennis racket, or even most smartphones (be careful not to break it though).
https://en.wikipedia.org/wiki/Tennis_racket_theorem
Personally, my favourite theorem in classical mechanics is the so called 'tennis racket theorem', sometimes known as the 'intermediate axis theorem'.
It explains why objects with roughly three different moments of intertia have unstable rotation about their intermediate moment.
It can be easily demonstrated with a tennis racket, or even most smartphones (be careful not to break it though).
https://en.wikipedia.org/wiki/Tennis_racket_theorem
There's a link on that wikipedia page to this youtube demo
which makes it really obvious
https://www.youtube.com/watch?v=4dqCQqI-Gis
https://www.youtube.com/watch?v=4dqCQqI-Gis
Here is a link to the unofficial html edition : https://tgvaughan.github.io/sicm/toc.html
I came to say how horrible the formatting of the original post is. But this site fixes that. Much better! Thanks!
I love this book! Prior to reading it, I had been getting confused when trying to learn classical mechanics. The book writes out everything explicitly in code, which let me use the software engineering part of my brain, and made everything easy to follow. Apparently I had been being held back by unfamiliarity with math formalisms!
If you're interested in the topic and know how to program, it might be worth a read in case it turns out you're in the same boat.
If you're interested in the topic and know how to program, it might be worth a read in case it turns out you're in the same boat.
Apparently written by the same person who wrote Structure and Interpretation of Computer Programs - Gerald Jay Sussman .
Who also wrote "Functional Differential Geometry": https://mitpress.mit.edu/books/functional-differential-geome...
Obligatory wikipedia link:
https://en.wikipedia.org/wiki/Gerald_Jay_Sussman
https://en.wikipedia.org/wiki/Gerald_Jay_Sussman
For the curious, 2015: https://news.ycombinator.com/item?id=9560567
2013: https://news.ycombinator.com/item?id=6947257
2010: https://news.ycombinator.com/item?id=1581696
2013: https://news.ycombinator.com/item?id=6947257
2010: https://news.ycombinator.com/item?id=1581696
Unfortunately, the code in this book uses a modified version of scheme which is no longer maintained.
There is a racket port; I do not know how complete it is, though a quick look at the open issues suggests not all the pieces are there yet: https://github.com/bennn/mechanics
The most recent version of `scmutils` I see is from 2016: https://groups.csail.mit.edu/mac/users/gjs/6946/linux-instal... and https://groups.csail.mit.edu/mac/users/gjs/6946/scmutils-tar...
Edit: There seems to be quite a few ports to various languages: https://github.com/search?q=scmutils No idea how many are fully featured/tested.
The most recent version of `scmutils` I see is from 2016: https://groups.csail.mit.edu/mac/users/gjs/6946/linux-instal... and https://groups.csail.mit.edu/mac/users/gjs/6946/scmutils-tar...
Edit: There seems to be quite a few ports to various languages: https://github.com/search?q=scmutils No idea how many are fully featured/tested.
Looks like MIT Scheme (on which the ScmUtils system runs) still gets some regular maintainence releases, though I don't know how active development is these days:
https://www.gnu.org/software/mit-scheme/ https://ftp.gnu.org/gnu/mit-scheme/stable.pkg/
Installation instructions for the ScmUtils package:
http://groups.csail.mit.edu/mac/users/gjs/6946/index.html
As far as I know, Gerry and Jack still teach the course every year (can someone currently at MIT verify?) and still use the system.
https://www.gnu.org/software/mit-scheme/ https://ftp.gnu.org/gnu/mit-scheme/stable.pkg/
Installation instructions for the ScmUtils package:
http://groups.csail.mit.edu/mac/users/gjs/6946/index.html
As far as I know, Gerry and Jack still teach the course every year (can someone currently at MIT verify?) and still use the system.
Yep! https://groups.csail.mit.edu/mac/users/gjs/6.945/
I'm in his group at MIT.
I'm in his group at MIT.
Good to hear! Please say hi to everyone. I used to hang out in the group back when it was still at Tech^2.
I had assumed it was being done in Python https://news.ycombinator.com/item?id=14167453
Nope, that's just the 'intro to programming' class. The class for this book is "Classical Mechanics: A Computational Approach" (I think) and it's definitely still taught in Scheme.
While I'm glad to hear that, how does that work given that the students presumably won't have been taught Scheme from SCIP beforehand? Is it just a given that students taking the class learned Scheme on their own?
Thanks, I had assumed the intro class changing would signal a comprehensive shift.
I maintain a port to Clojure https://github.com/littleredcomputer/sicmutils
I haven't done much with it recently but have plans in the back of my mind for when I have more free time.
I haven't done much with it recently but have plans in the back of my mind for when I have more free time.
Thanks, this is a nice port!
I worked through SICM using sicmutils as a backup. The MIT Scheme version sometimes "locked up" on my solutions to exercises and sicmutils did not (the Foucault pendulum problem comes to mind).
This video is an introduction to SICM and sicmutils:
https://www.youtube.com/watch?v=7PoajCqNKpg
I worked through SICM using sicmutils as a backup. The MIT Scheme version sometimes "locked up" on my solutions to exercises and sicmutils did not (the Foucault pendulum problem comes to mind).
This video is an introduction to SICM and sicmutils:
https://www.youtube.com/watch?v=7PoajCqNKpg
I'm so glad it was helpful.
We have some Jupyter notebooks, e.g. https://github.com/littleredcomputer/sicmutils/blob/master/j...
One on the Foucault pendulum sounds like it would make a nice addition. Perhaps it could be done in a way that wouldn't spoil the exercise.
We have some Jupyter notebooks, e.g. https://github.com/littleredcomputer/sicmutils/blob/master/j...
One on the Foucault pendulum sounds like it would make a nice addition. Perhaps it could be done in a way that wouldn't spoil the exercise.
I agree, the Foucault example would make a good Juypter demonstration. Showing the pendulum in 3D with a 2D projection (like a Spirograph) of the motion on the ground would be especially nice, as in:
https://demonstrations.wolfram.com/FoucaultsPendulum/
This might be possible in a Jupyter notebook with WebGL.
https://demonstrations.wolfram.com/FoucaultsPendulum/
This might be possible in a Jupyter notebook with WebGL.
Your video: https://www.youtube.com/watch?v=7PoajCqNKpg ?
that isn’t true. you can go install it now on linux, mac os x, or windows subsytem for linux, and it runs just fine. and it still gets some updates.
I haven't read beyond the first two chapters, but the clarity and expressiveness of the notation used is amazing.
I do wish more maths/physics etc would adopt similar, clearer, unambiguous notations.
I do wish more maths/physics etc would adopt similar, clearer, unambiguous notations.
What are the books you need to read, before this book, so it makes sense?
I'm an EE, but I admit I made no progress. Clearly I lack some deeper understanding - where can I best look to fix that?
I'm an EE, but I admit I made no progress. Clearly I lack some deeper understanding - where can I best look to fix that?
It's not a book that I would recommend but the lectures from a course: https://theoreticalminimum.com/courses/classical-mechanics/2...
This should give you plenty of intuition to then tackle SICM.
This should give you plenty of intuition to then tackle SICM.
I'll be honest, it's been a long time since I was studying classical mechanics, but 15 years ago I remember answering a similar question with "read the Feynman Lectures Volume I!" They are the edited version of the intro physics course he gave at CalTech... Here is a link: http://www.feynmanlectures.caltech.edu/I_toc.html
If you've not been previously exposed to stuff like Lagrangians and Hamiltonians, I'd suggest Taylor's Classical Mechanics text, which is quite friendly.
Excellent! Good companion to this is Richard Fitzpatrick's Computational Physics notes
http://farside.ph.utexas.edu/teaching/329/329.html
http://farside.ph.utexas.edu/teaching/329/329.html
[deleted]
> We hope others, especially our competitors, will adopt these methods, which enhance understanding while slowing research.
:)
:)
Is there a PDF-version?
Is there something similar for Quantum Mechanics? Both MIT and Harvard use Griffith's "Introduction to Quantum Mechanics," but it seems to emphasize computation and symbol manipulation over physical intuition.
For a compuatational intuition of quantum mechanics, I recomment picking up quantum computation!
Step 1: Pick up Nielsen and Chuang, the standard textbook in the field. Step 2: Solve the microsoft Q# Quantum Katas, which take you through basic quantum gates all the way up to quantum algorithms. Step 3: Bedtime read Scott Aaronson's "quantum computing since Democritus" for a great _view_ into the way a researcher in quantum computing thinks about quantum phenomenon, entanglement, and all that.
I _just_ finished a course in college that did this, and I gained a lot from it: A lot of it made me "feel" QM better than the physics courses ever did.
Step 1: Pick up Nielsen and Chuang, the standard textbook in the field. Step 2: Solve the microsoft Q# Quantum Katas, which take you through basic quantum gates all the way up to quantum algorithms. Step 3: Bedtime read Scott Aaronson's "quantum computing since Democritus" for a great _view_ into the way a researcher in quantum computing thinks about quantum phenomenon, entanglement, and all that.
I _just_ finished a course in college that did this, and I gained a lot from it: A lot of it made me "feel" QM better than the physics courses ever did.
There's [1], also by Sussman, which is "just" about differential forms and related math, and in the summary they refer to QM: "An explanation of the mathematics needed as a foundation for a deep understanding of general relativity or quantum field theory."
[1] https://mitpress.mit.edu/books/functional-differential-geome...
[1] https://mitpress.mit.edu/books/functional-differential-geome...
Having read through that it's not really directly relevant to quantum physics (at least what I learned in undergrad). It is super applicable to general relativity and electrodynamics though! It's a much shorter book than SICM but I still have yet to understand it through to the end. Tricky stuff, but fun to think about!
There's a book called "Structure and Interpretation of Quantum Mechanics", by R. I. G. Hughes which is split into two parts, the first describing the mathematical structure of QM and the second discussing philosophical interpretations. I haven't read "Structure and Interpretation of Classical Mechanics" so I don't know if it has anything in common besides the title, but I love SIQM. It's beautifully explained and approaches the subject from an unusual angle for an introductory book. It does have both these things in common with SICP, which was also co-written by Sussman.
I was reading old reviews of the two books here on HN a few days ago, and no, there's nothing in common besides the name.
If it was this review, it tells you nothing about the book: https//news.ycombinator.com/item?id=5706631. The Amazon reviews are a lot more informative.
That said, I agree that SIQM and SICP are totally different (again, I haven't read SICM), and I suspect the similar naming is a coincidence. In my mind, they're linked only by the name and and the qualities I mentioned before (they are two of my favourite books). I read SIQM as a physics undergraduate before I ever heard of SICP, and it has nothing whatsoever to do with computers. As a philosophy professor writing about quantum mechanics, I doubt the author was aware of the existence of SICP, let alone that he was attempting to "ride on its coattails" as the linked comment claims.
Regarding the names, SICP and SIQM use the terms "structure" and "interpretation" in very different ways. In SIQM it's structure in the sense of the mathematical structure of the theory (Hilbert spaces, operators, eigenvalues, etc) and interpretation in the sense of philosophical interpretations (eg. the Copenhagen interpretation. The title is therefore a straightforward description of the two separate sections of the book. With SICP I was never really clear on the meaning of the title, particularly the interpretation part: interpretation by a human or a computer, or is it deliberately ambiguous?
There's also a minor, but telling, difference in the titles: SIQM's title actually starts with "The".
That said, I agree that SIQM and SICP are totally different (again, I haven't read SICM), and I suspect the similar naming is a coincidence. In my mind, they're linked only by the name and and the qualities I mentioned before (they are two of my favourite books). I read SIQM as a physics undergraduate before I ever heard of SICP, and it has nothing whatsoever to do with computers. As a philosophy professor writing about quantum mechanics, I doubt the author was aware of the existence of SICP, let alone that he was attempting to "ride on its coattails" as the linked comment claims.
Regarding the names, SICP and SIQM use the terms "structure" and "interpretation" in very different ways. In SIQM it's structure in the sense of the mathematical structure of the theory (Hilbert spaces, operators, eigenvalues, etc) and interpretation in the sense of philosophical interpretations (eg. the Copenhagen interpretation. The title is therefore a straightforward description of the two separate sections of the book. With SICP I was never really clear on the meaning of the title, particularly the interpretation part: interpretation by a human or a computer, or is it deliberately ambiguous?
There's also a minor, but telling, difference in the titles: SIQM's title actually starts with "The".
What is physical intuition when you deal with quantum mechanics? Griffiths is a perfectly fine book, and if you want to go deeper perhaps Sukurai.
What is physical intuition when you deal with quantum mechanics?
Connecting the math with a physical scenario, and highlighting how your intuitions are wrong, to help you develop new ones.
Feynman's Lectures on Physics starts with the 2 slit experiment, then talks about shining light on the holes to see which hole the electron goes through, then making the energy (wavelength) of the light lower (longer) to try to disturb it less.
Or that you shouldn't think of filters as just removing parts of the wave. For example, put unpolarized light through a vertically polarizing filter, then a horizontally polarized one. No light comes out. Then, in between the two, put a filter at 45 degrees. Now, you do get some light coming out of that horizontal filter.
The math tells you this, of course, but if you don't have an intuitive sense of how that maps to the physical world, you can end up e.g. not noticing where there's a mistake in your calculation that would be obvious it you thought about what it means physically.
Connecting the math with a physical scenario, and highlighting how your intuitions are wrong, to help you develop new ones.
Feynman's Lectures on Physics starts with the 2 slit experiment, then talks about shining light on the holes to see which hole the electron goes through, then making the energy (wavelength) of the light lower (longer) to try to disturb it less.
Or that you shouldn't think of filters as just removing parts of the wave. For example, put unpolarized light through a vertically polarizing filter, then a horizontally polarized one. No light comes out. Then, in between the two, put a filter at 45 degrees. Now, you do get some light coming out of that horizontal filter.
The math tells you this, of course, but if you don't have an intuitive sense of how that maps to the physical world, you can end up e.g. not noticing where there's a mistake in your calculation that would be obvious it you thought about what it means physically.
“If you can’t explain it simply, you don’t understand it well enough,” followed by “For example, we write Lagrange’s equations in functional notation as follows:
D(∂2L ∘ Γ[q]) − ∂1L ∘ Γ[q] = 0”
Whoa Nellie! Gonna get bumpy.
Interesting -- I'm super rusty on my physics -- is there discussion of the exercises anywhere? It'd be fun to go through this and feel like I "got" it.
http://blog.jessriedel.com/2017/06/28/legendre-transform/