I, too, spent a long time staring at expressions like<p><pre><code> “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))”
</code></pre>
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<p><pre><code> Λ(x, p) = L(x, v(x, p))
∂₂Λ = ∂₂L ∂₂v
</code></pre>
so that you're not scratching your head about “why is the derivative of L with respect to <i>v</i> showing up here, v is now a function isn't it?”<p>The formulation of first f
derivatives as inverse functions is new to me but makes sense.<p>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?
The abuse of differentials in explanations like this reminds of this classic and insightful MathOverflow answer: <a href="https://math.stackexchange.com/questions/3266639/notation-for-partial-derivative-of-functions-of-functions" rel="nofollow">https://math.stackexchange.com/questions/3266639/notation-fo...</a>
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 <a href="https://en.wikipedia.org/wiki/Hamiltonian_mechanics#From_symplectic_geometry_to_Hamilton's_equations" rel="nofollow">https://en.wikipedia.org/wiki/Hamiltonian_mechanics#From_sym...</a> on Wikipedia has some details. This symplectic duality is my preferred way of looking at Hamiltonian-Lagrangian transitions.
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).<p>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).<p>While the observation is almost silly, it has very deep consequences for different characterizations of problems and other constructions!
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.<p>I feel like understanding the general convex conjugate and then seeing the Legendre transform as a special case is almost more intuitive.
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.
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).<p>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.<p>Also, side question, is the content of this post considered Tropical Mathematics?
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.<p>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<p>?
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.