A Legendre Dual is a mathematical transform which changes a function of one or more variables to a new function involving the derivatives of that function with respect to the replaced variable. They are of use in chemistry and physics, particularly in thermodynamics.

Basically, the transform allows us to express a function of multiple variable in a different form, replacing one of its variables with a derivative, while maintaining enough information to retrieve the original function by repeating the transform. The is primarily useful when constraints on a function are more easily expressed as constraints on one or more of that functions derivatives. Specifically, minimums and maximums are found when certain derivatives are 0. Often times when dealing with physical systems, specific quantities are not easily measured (like the entropy of a system) but various derivatives of those quantities are (like temperature which is ∂U/∂s.) Legendre Duals maintain the structure of a system/experiment while allowing us to change the variables we're looking at.

Starting with a function of two variables:

f(x,y)

We define the Legendre Dual of f with respect to x by a change to the variable u=∂f/∂x as:

g(u,y) = f(u,y) - ux

That is, g is a function of u (the partial derivative of x) and y (one of the original variables.)

This formulation of the Legendre Dual leads to some useful relations:

df = ∂f/∂x dx + ∂f/∂y dy
(those ∂f/∂x are partial derivatives)

for clarity of notation, let's define v = ∂f/∂y

df = u dx + v dy

From the definition of the Legendre Dual, we get:

dg = df - udx - xdu

dg = u dx + v dy - udx - xdu

dg = v dy - xdu

Which tells us that the partial derivative of g with respect to u (originally defined as the rate of change of x) is -x! All of the partial derivatives with respect to non-replaced variables remain the same.

We can also use this definition to prove that the Legendre Dual of g with respect to u returns us to the original function f.

Physically, the Legendre Dual usually reflects removing a constraint on a variable and replacing it with a constraint on the potential of the variable. For example, if we take a system of fixed volume and instead put it in contact with a piston under a constant pressure, it's easiest to shift from minimizing the internal energy U to minimizing it's dual with respect to volume (replacing volume with pressure = dU/dV.)

Most thermodynamic quantities of interest are Legendre Duals of a basic equation for potential energy, and as such are simple restatements of the idea that a system in equilibrium will have minimium energy.

U = U(S, V, N)

S is Entropy

V is Volume

N is the number of particles of a given species

The Legendre dual with respect to volume gives us the Enthalpy (since pressure(p) is dU/dV):

H = U(S,p,N) - pV

The Legendre dual with respect to entropy gives us the Helmholtz free energy (since temperature(t) is dU/dS):

F = U(t,V,N) - tS

If we transform both variables, we get the Gibbs free energy:

G = U(t,p,N) - pV - tS

These are usually expressed in differential form:
dH = dU - p dV
dF = dU - t dS
dG = dU - p dv - t dS

Lots of physical quantities are Legendre Duals of each other. The Hamiltonian and the Lagrangian are duals of each other, for example.

This is remembered from a thermodynamics class many years ago, I can't find a good treatment of Legendre Duals on the web. If anyone has a clearer treatment, or more examples, I'd be interested to hear about them.

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