[HN Gopher] Legendre transform, better explained (2017)
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Legendre transform, better explained (2017)
Author : harperlee
Score : 89 points
Date : 2024-04-03 11:44 UTC (11 hours ago)
(HTM) web link (blog.jessriedel.com)
(TXT) w3m dump (blog.jessriedel.com)
| cl3misch wrote:
| Previous comment thread:
| https://news.ycombinator.com/item?id=19765389
| metadat wrote:
| (2017)
| kevindamm wrote:
| Really enjoyed this post, it is both understandable and
| revelatory. I had some introduction to the concepts here from
| calculus and physics classes but my mathematics interests are
| more along the branches of Abstract Algebra than Analysis so I
| didn't expect to enjoy it much, and I wonder if I would have hung
| on for as long if it didn't have the poor exposition first (and
| the promise of a better presentation).
|
| I wonder if more math-related material was given in this "look
| how confusing, now wait look at it this way" would be more
| engaging, overall. Perhaps replacing the first part with a
| demonstration instead of mocking established representations. But
| maybe there is something to the "you're not alone, this way of
| looking at it is confusing and hand-wavy" even if done
| deliberately, just to give comfort to students making sense of a
| concept for the first time. Especially with math, I think mamy
| people would be more eager to learn it if that initial
| uncomfortable and confusing stage is considered normal for
| everyone.
|
| Also, side question, is the content of this post considered
| Tropical Mathematics?
| crdrost wrote:
| I, too, spent a long time staring at expressions like
| "half-invert p(x, v) to get v(x, p) s.t. p(x, v(x,
| q)) = q then the Legendre transform is
| H(x, p) = p v(x, p) - L(x, v(x, p))"
|
| And I did come to one of the same conclusions as this article,
| which is that if we're talking pure mathematics, these
| "thermodynamic" expressions like ([?]L/[?]v)_x, ([?]L/[?]p)_x are
| deeply easy to get confused about and in fact you should just say
| "the derivative of the function with respect to its first
| argument holding the other arguments constant" and therefore
| introduce different functions which compute the same value under
| different symbols, say L(x, p) = L(x, v(x, p))
| [?]2L = [?]2L [?]2v
|
| so that you're not scratching your head about "why is the
| derivative of L with respect to _v_ showing up here, v is now a
| function isn 't it?"
|
| The formulation of first f derivatives as inverse functions is
| new to me but makes sense.
|
| However, I do think that we do even worse with linear algebra. I
| believe I could walk up to any college senior in physics and they
| wouldn't know that "the determinant is the product of the
| eigenvalues," but this should be as well-known as "the
| mitochondria are the powerhouse of the cell." I think this is
| because we introduce a complicated way to calculate determinants
| and then we use determinants to calculate the eigenvalues?
| prof-dr-ir wrote:
| Agreed, the way thermodynamics is often taught is such a mess.
|
| My personal and controversial [0] take is that the free energy
| should really be seen as the Legendre transform of the entropy,
| not of the energy.
|
| I know it is ultimately semantics, but this viewpoint makes the
| passage from the micro-canonical to the canonical ensemble so
| much nicer. In particular, the saddle point approximation for
| the canonical partition function makes it natural that the
| ensembles are equivalent in the thermodynamic limit... through
| a Legendre transform!
|
| Bonus corollary: the statement mentioned in the blog about the
| derivatives being each other's inverses is just saying that
| T(E) and E(T) in respectively the micro-canonical and the
| canonical ensemble define the same relation between E and T.
|
| [0] Proof of controversiality: even Wikipedia disagrees with me
| here, see
| https://en.wikipedia.org/wiki/Thermodynamic_free_energy
| shiandow wrote:
| I'm not even sure if it makes sense to view it as a Legendre
| transform. Or well, it is one, I'm just not sure if it's a
| good _definition_.
|
| You get the free energy for 'free' if you use a Lagrange
| multiplier to maximize entropy while keeping the energy fixed
| (temperature is the inverse of that Lagrange parameter). In
| one fell swoop this shows why temperature is a thing and why
| minimizing the free energy is important.
|
| The Legendre transform just returns the value of the
| constraint from the minimized function, but at that point why
| bother?
|
| I do agree that it makes more sense to see the fee energy as
| a Legendre transform of the entropy, that's kind of what you
| end up doing if you minimize entropy in this way.
| shiandow wrote:
| I know programmers like to blame mathematicians for writing
| functions with lots of one letter variable names, but it's the
| physicists who insist on doing so without defining any of them.
|
| You want to know what V is? It's clearly the potential, we've
| defined it six papers ago! Oh you were wondering what it's type
| was, well it's usually a scalar field. No, don't write the
| parameter as t that changes the whole meaning!
| Phiwise_ wrote:
| Sussman has a great guest lecture that mentions exactly these
| sorts of issues, and that he found it much easier to verify
| work he and his grad students did in mathematical physics
| after developing the "mechanics programming" notation he
| explains in Structure and Interpretarion of Classical
| Mechanics and Functional Differential Geometry.
| mydogcanpurr wrote:
| > I think this is because we introduce a complicated way to
| calculate determinants and then we use determinants to
| calculate the eigenvalues?
|
| Yes, the determinant should be taught and defined as the volume
| of the parallelepiped in n-dimensions defined by the columns of
| the given square matrix. This perspective makes it immediately
| obvious that the eigenvalues scale the parallelepiped in each
| of its dimensions (a basis of eigenvectors makes it even
| simpler). Of course the volume (determinant) must be the
| product of these scaling factors (eigenvalues)! Since algebra
| is too convenient for solving problems, this geometric
| intuition is often an afterthought if it's even taught at all.
| lupire wrote:
| What trash math classes were you all in that didn't teach all
| of this?
| ericdfoley wrote:
| Helliwell & Sahakian Modern Classical Mechanics at least seems to
| do a much better job of explaining the Legendre transform than
| Goldstein, but it still never mentions the convexity requirement
| on f.
|
| I feel like understanding the general convex conjugate and then
| seeing the Legendre transform as a special case is almost more
| intuitive.
| ykonstant wrote:
| Very nice article, kudos to the author. The inverse relation
| between Jacobians generalizes to a duality statement via the
| symplectic structure of the configuration space; the section
| https://en.wikipedia.org/wiki/Hamiltonian_mechanics#From_sym...
| on Wikipedia has some details. This symplectic duality is my
| preferred way of looking at Hamiltonian-Lagrangian transitions.
| doppioandante wrote:
| Wow, I've been looking for a meaningful definition of the
| Legendre transform for ages, thanks for writing this up
| LolWolf wrote:
| It's neat! To be fair, as a physicist, I did not understand the
| Legrendre transform essentially until taking convex optimization
| (where it is known as the Fenchel conjugate).
|
| Many sources, but all of them are reasonable and give a
| constructive definition that actually explains what it does: we
| can characterize a function either by its graph, or its
| supporting hyperplanes (when it is a closed, convex function).
|
| While the observation is almost silly, it has very deep
| consequences for different characterizations of problems and
| other constructions!
| bigbacaloa wrote:
| I found this explanation quite bad. Poorly motivated and making a
| priori regularity assumptions that are not necessary. The quoted
| explanation by Arnold is much better.
| ijustlovemath wrote:
| The abuse of differentials in explanations like this reminds of
| this classic and insightful MathOverflow answer:
| https://math.stackexchange.com/questions/3266639/notation-fo...
| abetusk wrote:
| Thank you so much for this link. I was having trouble following
| some of the notation that came up with automatic
| differentiation and I think this clears it up.
| dang wrote:
| Thanks! That is enlightening. It made me realize that the
| confusion I always felt was actually in the notation all along.
|
| That link was discussed here btw:
|
| _On Leibniz Notation_ -
| https://news.ycombinator.com/item?id=39064174 - Jan 2024 (95
| comments)
| SpaceManNabs wrote:
| How beautiful that this blog post was made years after I was
| struggling in thermodynamics to understand these transforms. Now
| if someone could make a post for the Laplace Transform with the
| audience being people familiar with the fourier transform.
| mgaunard wrote:
| Not nearly as good as the explanation of the Fourier transform we
| had the other day.
| lupire wrote:
| Legendre transform moves a ruler (tangent line) along a convex
| function, measuring how much "lag" the function accumulated while
| "accelerating" up to its velocity at a certain moment, relative
| to having constant velocity for its entire history.
|
| The larger the function's 2nd derivative is, the smaller the
| transform value is. And vice versa. Since the tranform is written
| in terms of the original function's derivative, not it's "x
| value", the derivative of the transform is inversely proportional
| to the derivative of the function
|
| ?
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