       Legendre transform

• To: mathgroup at smc.vnet.net
• Subject: [mg64312] Legendre transform
• From: Tobin Fricke <fricke at ocf.berkeley.edu>
• Date: Fri, 10 Feb 2006 03:30:08 -0500 (EST)
• Organization: University of California, Berkeley
• Sender: owner-wri-mathgroup at wolfram.com

```As an exercise in learning Mathematica, and also as a computational and
pedagogical tool for myself, I'm attempting to implement a function to
compute the Legedre transform of an expression.

The procedure is, given a function f(x):

1. compute y(x) = f'(x) and solve for x(y)
2. the legendre transform is g(y) = -f(x(y)) + x(y) y

Here's my first attempt:

legendreTransform[f_, x_, y_] := Block[{},
solutions = Solve[y == D[f, x], x];
x[k_] := x /. First[solutions];
-(f /. x -> x[y]) + x[y] y]

For instance, it correctly gives the transform of xLog[x]-x as e^x:

Simplify[legendreTransform[x Log[x] - x, x, y],
Assumptions -> {Element[y, Reals]}]

This seems to work, though (1) things go horribly wrong if x==y, and (2)
errors aren't handled at all--for instance, it should detect whether the
Solve[] fails, etc.

Another attempt is written in a such a way as to operate on *functions*
rather than expressions.  To me this seems like the preferred way, but
maybe it isn't:

legendreTransform[f_] = Block[{x, y},
x[y_] = x /. First[Solve[y == f'[x], x]];
Function[y, -f[x[y]] + x[y] y]]

Any hints/critiques/etc appreciated.  I'd like to end up with a version
that will let me take the legendreTransform with respect to an arbitrary
argument of a function with arbitrarily many arguments.

thanks,
Tobin

 http://en.wikipedia.org/wiki/Legendre_transform#Legendre_transformation_in_one_dimension

```

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