2.4.2 Legendre transformation

Consider a function $f:\mathbb{R}\to\mathbb{R}$ , whose argument we denote by $\xi$ (the curve in Figure 2.10 is a possible graph of $f$ ). For simplicity we are considering the scalar case, but the extension to $f:\mathbb{R}^n\to\mathbb{R}$ is straightforward. The Legendre transform of $f$ will be a new function, $f^*$ , of a new variable, $p\in\mathbb{R}$ .

Figure: Legendre transformation

Let $p$ be given. Draw a line through the origin with slope $p$ . Take a point $\xi=\xi(p)$ at which the (directed) vertical distance from the graph of $f$ to this line is maximized:

$$\xi(p):=\arg\max_\xi\{p\xi-f(\xi)\}.$$ (2.33)

(Note that $\xi(p)$ may not exist, so the domain of $f^*$ is not known a priori. Also, $\xi(p)$ is not necessarily unique unless $f$ is a strictly convex function.) Now, define $f^*(p)$ to be this maximal value of the gap between $p\xi$ and $f(\xi)$ :

$$f^*(p):=p\xi(p)-f(\xi(p))=\max_\xi\{p\xi-f(\xi)\}.$$ (2.34)

We can also write this definition more symmetrically as

$$f^*(p)+f(\xi)=p\xi$$ (2.35)

where $p$ and $\xi=\xi(p)$ are related via (2.34). When $f$ is differentiable, the maximization condition (2.34) implies that the derivative of $p\xi-f(\xi)$ with respect to $\xi$ must equal 0 at $\xi(p)$ :

$$p-f'(\xi(p))=0.$$ (2.36)

Geometrically, the tangent line to the graph of $f$ at $\xi(p)$ must have slope $p$ , i.e., it must be parallel to the original line through the origin (see Figure 2.10). If $f$ is convex then (2.34) and (2.37) are equivalent.

The Legendre transformation has some nice properties. For example, $f^*$ is a convex function even if $f$ is not convex. The reason is that $f^*$ is a pointwise maximum of functions that are affine in $p$ , as is clear from (2.35). Also, for convex functions the Legendre transformation is involutive: if $f$ is convex, then $f^{**}=f$ .

Now let us return to the Hamiltonian $H$ defined in (2.29). We claim that it can be obtained by applying the Legendre transformation to the Lagrangian $L$ . More precisely, for arbitrary fixed $x$ and $y$ let us consider $L(x,y,y')$ as a function of $\xi=y'$ . The relation (2.37) between $p$ and $\xi(p)=y'(p)$ becomes

$$p-{L}_{ y'}(x,y,y'(p))=0$$ (2.37)

which corresponds to our earlier definition (2.28) of the momentum $p$ . Next, (2.35) gives

$$L^*(x,y,p)=py'(p)-L(x,y,y'(p))$$ (2.38)

which is essentially our earlier definition (2.29) of the Hamiltonian $H$ . But there is a difference: in (2.29) we had $y'$ as an independent argument of $H$ , while in (2.39) $y'$ is a dependent variable expressed in terms of $x,y,p$ by the implicit relation (2.38). In other words, the Legendre transform of $L(x,y,y')$ as a function of $y'$ (with $x,y$ fixed) is $H(x,y,p)$ , which is a function of $p$ (with $x,y$ fixed) and no longer has $y'$ as an argument. Note that the above derivation is formal, i.e., we are ignoring the question of whether or not (2.38) can indeed be solved for $y'$ . This issue did not arise earlier when we were working with the Hamiltonian $H=H(x,y,y',p)$ .

The above approach has another, more important drawback. Recall the observation based on (2.31) that $H$ has a stationary point as a function of $y'$ along an optimal curve. This property will be crucial later; combined with the canonical equations (2.30), it will lead us to the maximum principle. But it only makes sense when we treat $y'$ as an independent variable in the definition of $H$ . On the other hand, Hamilton and other 19th century mathematicians did not write the Hamiltonian in this way; they followed the convention of viewing $y'$ as a dependent variable defined implicitly by (2.38). This is probably why it was not until the late 1950s that the maximum principle was discovered.