# Physical meaning of Legendre transformation

I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and mechanics and it seemed only a change of coordinates?

Thanks.

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See

http://en.wikipedia.org/wiki/Legendre_transformation#Applications

In theoretical physics, the basic or defining mathematical properties of the Legendre transformation are used to switch between one form of the energy - or "potential", as the generalized energies are called in thermodynamics - to another.

This is important to switch between the Lagrangian in abstract mechanics that depends on $x,v$ (positions and velocities) to the Hamiltonian, the true energy that depends on $x,p$.

In thermodynamics, the number of applications and "types of switches" is even higher. You may go from energy to enthalpy or Helmholtz free energy or Gibbs free energy by Legendre-transforming with respect to various variables. The transform goes back and forth. As the Wikipedia example explains, there are other useful variables that you may Legendre-transform with respect to, including the charge and voltage.

You may consider the Legendre transformation to be a "mere" redefinition of variables - but that's why it's so important in practice. In reality, the different ways to describe the system that differ by a Legendre transformation are "equally fundamental" or "equally natural" so it's often useful to be familiar with all of them and to know what is the relationship between them. The relationship is given by the Legendre transformation.

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Hi Luboš, I had to remove the greeting and signature in accordance with meta.physics.stackexchange.com/q/360/365#365 (I don't know whether you've seen that yet). Since Jeff has handed down the official word on greetings and signatures from the SE team, we (mods at least) have to be kind of strict about making these edits. It'd be helpful if you could omit those from future posts. – David Zaslavsky Feb 1 '11 at 18:11
Dear David, sorry for that. I've erased a few greetings from some of my other answers. It's a polite rhythm I have simply gotten used to and it's somewhat hard to unlearn it. – Luboš Motl Feb 1 '11 at 18:44
-1: "equally fundamental" or "equally natural" are any reversible variable changes. Practical are those that help separate new variables and/or cast the equations in a "solvable" form. If a problem is not resolved in the Lagrangian form, it remains such in the Hamilton form. – Vladimir Kalitvianski Feb 1 '11 at 20:50
@Vladimir: they are all equally fundamental in that the formulations are completely equivalent (of course, the usual conditions on convexity, etc. have to be satisfied). So this lets you decide which formulation is the best one for the given problem. But this is precisely what Luboš said too so I don't get what you disagree with... – Marek Feb 2 '11 at 13:40
Thanks, very enlightening your answer! – gsAllan Feb 2 '11 at 14:25
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 @ Vladimir, Thanks for the reference, I'll read! – gsAllan Feb 2 '11 at 14:26

It isn't "physical" intuition per-se, but I find the convex-analysis interpretation of the Legendre transform to be the most enlightening.

A convex set is uniquely determined by it's supporting hyperplanes, and the Legendre transform is an encoding of the convex hull of a function's epigraph in terms of it's supporting hyperplanes. If the function is convex and differentiable, then the supporting hyperplanes correspond to the derivative at each point, so the Legendre transform is a reencoding of a function's information in terms of it's derivative.

Here are a few links that illustrate the concept:

http://jmanton.wordpress.com/2010/11/21/introduction-to-the-legendre-transform/ (great in-depth explanation)

http://www.mia.uni-saarland.de/Teaching/NAIA07/naia07_h3_slides.pdf (compuational perspective)

http://maze5.net/?page_id=733 (graphical/visual explanation on my website. Not that advanced, but but with a lot of pictures)

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