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Sussman is one of the very few classical mechanics textbooks that gives a reasonable definition of the Legendre transform. Most physicists cannot actually tell

Sussman is one of the very few classical mechanics textbooks that gives a reasonable definition of the Legendre transform. Most physicists cannot actually tell you what that transform is, even though it sits at the heart of both classical and quantum mechanics. http://blog.jessriedel.com/2017/06/28/legendre-transform/ http://blog.jessriedel.com/2017/06/28/legendre-transform/

jules 8y ago
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
- jessriedel 8y ago
  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.
  - Misdicorl 8y ago
    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.
- deleted 8y ago
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- bollu 8y ago
  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 :)
  - jules 8y ago
    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.
ananduri 8y ago
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.
chickenmonkey 8y ago
As a college sophomore majoring in Physics, I'm curious: what level of student is this text accessible to?
- thatcherc 8y ago
  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!
  - noobly 8y ago
    Was this book your first exposure to Classical Mechanics? Would you recommend it as such?
    - thatcherc 8y ago
      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!
archgoon 8y ago
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.
- petschge 8y ago
  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.
  - jessriedel 8y ago
    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.
martincmartin 8y ago
What other textbooks would you suggest?
- jessriedel 8y ago
  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.
  - eternauta3k 8y ago
    > No good textbooks on quantum mechanics or QFT exist No love for Cohen-Tannoudji?
    - jessriedel 8y ago
      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-quantum-mechanics-part-0-measurements-are-about-bases/ http://blog.jessriedel.com/2016/11/12/how-to-think-about-qua...
  - jagiammona 8y ago
    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.
hansen 8y ago
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
- jessriedel 8y ago
  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.
- fspeech 8y ago
  What is a good reference from the symplectic geometry angle?
  - hansen 8y ago
    The standard (but rather heavy and difficult) is Abraham, Marsden: Foundations of Mechanics More gentle is Vladimir Arnold: Mathematical Methods of Classical Mechanics
indweller 8y ago
Your blog is a great resource. Especially the fact that you've linked other blogs and links is really helpful.
leephillips 8y ago
This is a wonderful article. I'm sure I never really understood what the Legendre transformation was before - not this well.
xtacy 8y ago
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 https://www.youtube.com/watch?v=GOkZs2RZMQY.
tagrun 8y ago
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).
- jessriedel 8y ago
  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.

jules 8y ago

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

jessriedel 8y ago
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.
- Misdicorl 8y ago
  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.
deleted 8y ago
[deleted]
bollu 8y ago
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 :)
- jules 8y ago
  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.

ananduri 8y ago

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.

chickenmonkey 8y ago

As a college sophomore majoring in Physics, I'm curious: what level of student is this text accessible to?

thatcherc 8y ago
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!
- noobly 8y ago
  Was this book your first exposure to Classical Mechanics? Would you recommend it as such?
  - thatcherc 8y ago
    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!

archgoon 8y ago

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.

petschge 8y ago
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.
- jessriedel 8y ago
  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.

martincmartin 8y ago

What other textbooks would you suggest?

jessriedel 8y ago
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.
- eternauta3k 8y ago
  > No good textbooks on quantum mechanics or QFT exist No love for Cohen-Tannoudji?
  - jessriedel 8y ago
    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-quantum-mechanics-part-0-measurements-are-about-bases/ http://blog.jessriedel.com/2016/11/12/how-to-think-about-qua...
- jagiammona 8y ago
  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.

hansen 8y ago

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

jessriedel 8y ago
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.
fspeech 8y ago
What is a good reference from the symplectic geometry angle?
- hansen 8y ago
  The standard (but rather heavy and difficult) is Abraham, Marsden: Foundations of Mechanics More gentle is Vladimir Arnold: Mathematical Methods of Classical Mechanics

indweller 8y ago

Your blog is a great resource. Especially the fact that you've linked other blogs and links is really helpful.

leephillips 8y ago

This is a wonderful article. I'm sure I never really understood what the Legendre transformation was before - not this well.

xtacy 8y ago

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 https://www.youtube.com/watch?v=GOkZs2RZMQY.

tagrun 8y ago

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).

jessriedel 8y ago
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.