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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:

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

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

We can also write this definition more symmetrically as

$\displaystyle f^*(p)+f(\xi)=p\xi$ (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)$ :

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

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

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

$\displaystyle L^*(x,y,p)=py'(p)-L(x,y,y'(p))$ (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.

next up previous contents index
Next: 2.4.3 Principle of least Up: 2.4 Hamiltonian formalism and Previous: 2.4.1 Hamilton's canonical equations   Contents   Index
Daniel 2010-12-20