Lagrange multiplier applied to the Legendre transform

Since the Legendre transform is a stationary point of a function, Lagrange multipliers should be the natural choice for handling constraints. However, there is a problem due to the fact that we are mostly interested in its functional dependence on the transform parameter, and not on its value.

THE LEGENDRE-LAGRANGE PROBLEM

Let $f(\overline{x})$ be a function of the real n-vector $\overline{x}$ and $g(\overline{p})$ its Legendre transform defined by

$g(\overline{p})=\overline{p}.\overline{x}-f(\overline{x})\mathit{\hspace{1em}\hspace{1em}}{p}_i={\displaystyle \frac{\partial f}{\partial x_i}}$(1)

The vector $\overline{p}$ is a function of $\overline{x}$, or conversely, $\overline{x}$ is a function of $\overline{p}$, so that $g(\overline{p})$ is a function of $\overline{p}$ alone.The Legendre transform is also alternatively defined as the maximum of the function $\overline{p}.\overline{x}-f(\overline{x})$. This maximum is reached for $\overline{x}$ satifying the second set of equations in (1).Suppose now that the components $x_i$ are not all independent, but are linked by the constraint:

$h(\overline{x})=\mathrm{\hspace{0.33em}0}$(2)

This equation defines one of the components, $x_{\alpha }$ for example, as a function of all the others. Putting this function into (1) would give us the transform $g(\overline{p})$ as a function of the vector $\overline{p}$ with n-1 components. It would be nice if we could instead use a Lagrange multiplier $k$ and compute the maximum of the function

$$\overline{p}.\overline{x}-f(\overline{x})+kh(\overline{x})$$

with all its n components. But then, how does the constraint (2) on $\overline{x}$ translate to $\overline{p}$? The answer is amazingly simple, as we shall see next.

DIRECT COMPUTATION OF THE TRANSFORM

We are going first to compute the Legendre transform the hard way, without the help of a multiplier, by considering $x_{\alpha }$ as a function of the other components:

$p_i={\displaystyle \frac{df}{d{x}_i}}={\displaystyle \frac{\partial f}{\partial x_i}}+{\displaystyle \frac{\partial f}{\partial x_{\alpha }}}{\displaystyle \frac{\partial x_{\alpha }}{\partial x_i}}\mathit{\hspace{1em}\hspace{1em}}{\displaystyle \frac{dh}{d{x}_i}}={\displaystyle \frac{\partial h}{\partial x_i}}+{\displaystyle \frac{\partial h}{\partial x_{\alpha }}}{\displaystyle \frac{\partial x_{\alpha }}{\partial x_i}}=0\mathit{\hspace{1em}\hspace{1em}}i\ne \alpha$(3)

Taking the value of $\frac{\partial x_{\alpha }}{\partial x_i}$ from the second set of equations (3) and putting it into the first set, we get:

$p_i={\displaystyle \frac{\partial f}{\partial x_i}}-\mathrm{\Phi }{\displaystyle \frac{\partial h}{\partial x_i}}\mathit{\hspace{1em}}\text{with}\mathit{\hspace{1em}}\mathrm{\Phi }={\displaystyle \frac{\frac{\partial f}{\partial x_{\alpha }}}{\frac{\partial h}{\partial x_{\alpha }}}}\mathit{\hspace{1em}}\text{and}\mathit{\hspace{1em}}i\ne \alpha$(4)

From definition (1), the Legendre transform $g$ is therefore:

$g(\overline{p})={\displaystyle \sum _{i\ne \alpha }}{x}_i\left({\displaystyle \frac{\partial f}{\partial x_i}}-\mathrm{\Phi }{\displaystyle \frac{\partial h}{\partial x_i}}\right)-f(\overline{x})$(5)

BACK TO THE LAGRANGE MULTIPLIER

In the first equation (4), if we set $i=\alpha$ we get $p_{\alpha }=0$ and conversely, getting the value of $\mathrm{\Phi }$. So, in (5), we may remove the condition $i\ne \alpha$ and sum over all the n components of $\overline{p}$. The constraint $p_{\alpha }=0$ reduces the number of independent components to n-1. In fact, we are back to the traditional Lagrange method with the multiplier $\mathrm{\Phi }$ and an additional constraint. We even have the choice between n such constraints. They generate up to n functionally different transforms corresponding to the n possible forms of the function $\overline{x}$, according to which component $x_{\alpha }$ we choose to eliminate.The method extends easily to the case of more than one constraint:
$h_1=h_2=\mathrm{\dots }{h}_m=0$. We use m multipliers ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2\mathrm{\dots }{\mathrm{\Phi }}_m$. They are computed by equating to zero any set of m components from $\overline{p}$.