Legendre transform, better explained (2017)

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?

Re: Legendre transform, better explained (2017)

#6

Post by crdrost » Wed, Apr 03, 2024, 3:42 PM UTC

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

    Λ(x, p) = L(x, v(x, p))
    ∂₂Λ = ∂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?

Re: Legendre transform, better explained (2017)

#7

Post by ericdfoley » Wed, Apr 03, 2024, 3:52 PM UTC

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.

Re: Legendre transform, better explained (2017)

#8

Post by ykonstant » Wed, Apr 03, 2024, 3:56 PM UTC

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.

Re: Legendre transform, better explained (2017)

#9

Post by prof-dr-ir » Wed, Apr 03, 2024, 4:01 PM UTC

crdrost wrote: post #6

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 s…

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

Re: Legendre transform, better explained (2017)

#10

Post by doppioandante » Wed, Apr 03, 2024, 4:09 PM UTC

Wow, I've been looking for a meaningful definition of the Legendre transform for ages, thanks for writing this up