The term "transformation" is often used in physics and mathematics for functions to denote a different (and useful) way of encoding the information in a function. In this question, I want to focus on that way of looking at it. There are two features we care about then:
- The transformation must preserve information. Equivalently, any transformation must be invertible.
- The transformed function should recast the information contained in the original function in a useful way.
For example, roughly speaking, the Fourier transformation transforms the information in a function from an infinite set of positions and amplitudes to an infinite set of sine-waves and the amplitude of those sine-waves.
It is not necessarily obvious that this should be possible to do in an information preserving way. But, it can be, and the usefulness is clear: sine-waves are easy to work with, form a basis, and many physical systems (at least to first order) transmit sine-waves without any change to that individual sine wave.
The Legendre transformation seems less clear. Let us consider one of the graphical ways of looking at the Legendre transformation. The information of our original function $f(x)$ is an infinite set of positions versus amplitudes. The Legendre transformation has the information represented in a different way: the derivative (slope) of the original function, $p$, versus the intersection with the $y$-axis.
It is clear that having a new information-representation using the slope $p$ is potentially interesting, but why is the slope versus intersection with the $y$-axis interesting?
We could, instead, for example, create a transformation which is the original function $f(x),$ but represented as a function of the slope $f(p)$ - we would just need to remember the $y$ value of a single point, let's say $f(x=0).$
In thermodynamics one sees that minimums of different potentials are useful in different circumstances, and it is not too difficult to demonstrate this mathematically. But from the graphical point of view, why should the point at which minimum value of the (slope, $y$-axis) is taken on seem interesting, a priori?
A note about other questions on this website:
This is a related question that has several answers:
- This answer demonstrates mathematically that Legendre transformations appear naturally in thermodynamics, but does not refer to any "graphical" intuition.
- This neatly shows how and why the Legendre transformation is information preserving, but does little to answer the question of why it is specifically the "y-axis" that would be interesting.
- This answer again shows why the Legendre transformation works.
- This answer emphasizes that switching to an information description in terms of the slope could be useful, but again does not explain why it is the slope versus intersection with the $y$-axis that is interesting.
This question is much more closely related to the spirit of my question, and has received no satisfactory answers.
This pedagogical paper is very close to the spirit of my question. It claims that the significance of the intersection with the $y$-axis is the fact that this choice makes the Legendre transformation an involution. While this is certainly interesting it still feels unsatisfying. What physical interpretation can the Legendre transformation be given?
