Talk:Legendre transformation

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The abrupt way this article begins with no context-setting and no clear definition should grate upon the sensibilities of thoughtful people. Those who know this topic, please de-stubbify. Michael Hardy 22:13, 4 Feb 2004 (UTC)

Contents

[edit] Huh?

This is absolutely useless as explanation of Legendre transformation. It's not even a definition, let alone explanation.

I'm inclined to agree. I don't know the subject, but I know enough to know that whoever wrote this did not write clearly. <POV> Physicists often seem opposed to writing clearly about mathematics; they prefer a touchy-feely style. </POV> Could the author of the above identify "themself"? Thanks. Michael Hardy 22:14, 10 Sep 2004 (UTC)
<POV>Mathematicians often write like Bourbaki and write incomprehensible abstract stuff at a far higher level of abstraction and generality than needed for practical applications and are ever ready to criticize physicists for being "nonrigorous" and "intuitive".</POV>—the preceding unsigned comment is by Tweet Tweet (talk • contribs) 06:53, 12 November 2004 (UTC1)
<POV> Mathematicians sometimes write in more generality than is needed for a particular application, but greater abstraction can help make many things clearer and illuminate similarities between different subjects (e.g. the Hamiltonian's use in both classical and quantum mechanics). If mathematicians claim that physicists are not rigorous, it is only because they make statements that are not necessarily clear and obvious or not necessarily true. That said, there is usually compelling reason to give both clear and precise mathematical definitions and explanations/examples to illustrate what the definitions mean. </POV> 75.3.116.185 06:15, 12 June 2007 (UTC)

[edit] Possible copyright problem

The exposition quite closely follows the reference Rockafellar. I hope this is not a copyright problem. —the preceding unsigned comment is by 84.239.128.9 (talk • contribs) 09:26, 31 October 2005 (UTC1)

[edit] Possible clarifications/improvements

What does this mean:

f(x) + g(y) = \left\langle x,y\right\rangle.

?? Specifically, what is the "inner product"-looking thing? —the preceding unsigned comment is by TobinFricke (talk • contribs) 20:39, 22 July 2005 (UTC1)

I'm new to all this stuff here. To editing wikipedia as well as of Legendre transformation. But I think this article contains much information -- compared with the german, italian or slovenian page. To clarify it, one should perhaps begin with:

Given are two open subsets U and V of Rn and two real-valued differentiable functions f and g such that the first derivative Df is a bijection (one-to-one correspondence) U→V and Dg is a bijection V→U.
Then, f and g are said to be Legendre transforms of each other if
Df = \left( Dg \right)^{-1}.

This can still be clarified by explaining how Df must be interpreted as a function U→V, if this is necessary: if x = (x1, ..., xn) is in U and y = (y1, ..., yn), then y = Df(x) means:

y_i = {\part f \over \part x_i}

In the following lines, I would replace the confusing expression

f(x) + g(y) = \left\langle x,y\right\rangle.

by

f(x) + g(y) = \sum_{i=1}^{n}x_{i} \cdot y_{i}

In the german article it is said that the Legendre transforms is a special case of the "Berührungstransformation". I don't know what this is, it is not explained in further details. But perhaps "Berührungstransformation" means: Only the condition Df = \left( Dg \right)^{-1} holds?

I hope, my suggestions are not too confusing.

-- Mathias Michaelis 12:30, 2005 Oct 25 (MET)

The english term for Berührungstransformation would be contact transformation or contactomorphism, which denotes mappings between manifolds that preserve contact structures. Okay, that probably is not too helpful... Maybe a quick look at contact geometry or the page about contact manifolds at PlanetMath helps. That being said: be bold! What you described above would certainly increase the value of this entry. —Tobias Bergemann 09:54, 23 November 2005 (UTC)

What happens whenever Legendre transform is zero?..then most of the properties here couldn't e applied and

Df = \left( Dg \right)^{-1}.

would make no sense. --217.130.79.88 12:19, 13 November 2006 (UTC)

The Legendre transform only applies to convex functions (d2f / dx2 > 0). f(x)=0 is not convex. PAR 17:23, 13 November 2006 (UTC)

[edit] Convex conjugation

Should the section on convex conjugates be moved to a separate page? -- Tobias Bergemann 10:54, 2004 Nov 4 (UTC)

I have moved the section on convex conjugation to its own page. —Tobias Bergemann 09:30, 23 November 2005 (UTC)

[edit] Fenchel duality

Among other things, Fenchel's theorem, Fenchel's duality theorem and Fenchel duality were redirected to Legendre transformation. However, I don't remember this page ever having any content about Fenchel duality. Did this content somehow get lost in the wiki history of this page? —Tobias Bergemann 09:35, 23 November 2005 (UTC)

[edit] Name for this property?

Does anybody know how the property

f(x_1, \dots , x_N) = g(x_1) + \cdots + g(x_N) \; \Rightarrow \; f^\star(p_1, \dots , p_N) = g^\star(p_1) + \cdots + g^\star(p_N)

is refered to in the literature? —the preceding unsigned comment is by Tobias Bergemann (talk • contribs) 12:12, 13 December 2005 (UTC1)

[edit] application of Legendre Transform in thermodynamics

The very idea that ENTHALPY H could be something as a Legendre Transform of the internal energy U is without any scientific evidence. The function (U+pV) was introduced by Rankine in 1854, and named ENTHALPY in 1922. Only in 1960 Callen and Tiszla formulated the hypothesis that enthalpy could be viewed as a Legendre Transform : before that enthalpy was a simple energy-variable ("heat-content at constant pressure"). Any-one interested can get a file with more info on this. —the preceding unsigned comment is by Smannaerts (talk • contribs) 22:28, 4 January 2006 (UTC1)


The idea that enthalpy is a Legendre transform of the internal energy is a mathematically provable fact. Callen and Tiszla did not hypothesize, they recognized that this is true. PAR 05:53, 7 November 2006 (UTC)