The Legendre transform of a function is a new function , which depends only on the derivative of , and from which the original can be reconstructed. The point is that contains the same information as , just in a different form, analogously to e.g. the Fourier transform.
Let us choose an arbitrary point in the domain of . Consider a line tangent to at , which has slope and intersects the -axis at :
Where . We now define the Legendre transform , such that for all we have (some authors use instead). Renaming to :
We want this function to depend only on the derivative , but currently still appears here as a variable. We solve this problem in the easiest possible way: by assuming that is invertible for all . If is the inverse of , then is given by:
The only requirement for the existence of the Legendre transform is thus the invertibility of in the target interval , which is only satisfied if is either convex or concave, meaning its derivative is monotonic.
The derivative of with respect to is simply . In other words, the roles of and are switched by the transformation: the coordinate becomes the derivative and vice versa:
Furthermore, Legendre transformation is an involution, meaning it is its own inverse. To show this, let be the Legendre transform of :
Moreover, a Legendre transform is always invertible, because the transform of a convex function is itself convex. Convexity of means that for all , so a proof is:
And an analogous proof exists for concave functions where .
- H. Gould, J. Tobochnik, Statistical and thermal physics, 2nd edition, Princeton.