What properties, specifically, make the Legendre transform so useful in physics?

The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e.g. Physical meaning of Legendre transformation, Why is the Hamiltonian the Legendre transform of the Lagrangian?, Equivalence between Hamiltonian and Lagrangian Mechanics, as well as many others).

I am not asking how the Legendre transform is defined, or how it works to connect Lagrangian and Hamiltonian formalisms, so let's take all of this for granted here.

(The main question) One thing that always bugged me is: why specifically is the Legendre transform so useful? Starting from a Lagrangian viewpoint, principle of least action and all that, one typically defines the Hamiltonian via Legendre transform and shows that this new formalism is oh so useful for a myriad of reasons. But why does this happen? What is it about the Legendre transform that makes this possible?

(Geometrical viewpoint) The Legendre transform has a pretty clear geometrical interpretation in the context of convex analysis: it can be used to switch to a "dual" description of a convex set in terms of its supporting hyperplanes (see e.g. the nice description of how this works in this answer, or any introductory book about convex geometry). Lagrangian and Hamiltonian are also often understood in (differential) geometric terms: one can think of the Lagrangian as a functional on the tangent bundle of some underlying differential manifold, $$L:TM\to\mathbb R$$, while the Hamiltonian is a functional on the cotangent bundle, $$H:T^*M\to\mathbb R$$.

Are these geometrical interpretations related in any way? Is there merit in thinking the Hamiltonian as related to the Lagrangian in a similar fashion as how one can describe convex sets in terms of their supporting hyperplanes? I suppose to some degree this is trivially true: $$H$$ being the Legendre transform of $$L$$ with respect to the $$\dot q,p$$ parameters means (I think) that some sections of the epigraphs of $$L$$ and $$H$$ are related in this way. But that still doesn't tell me why such a dual description of sections of the epigraph of the Lagrangian should be of physical interest..

(Functional viewpoint) Another possible route to an answer might be in asking about the properties of the transform that make it useful. In other words, if I were to start from the Lagrangian formalism, and consider different possible functional transforms $$L\mapsto G[L]$$, would I get to the conclusion that $$G=(\text{Legendre transform})$$ is a good choice by asking for some specific property I want in the new formalism? Are there formalisms other than the Lagrangian and Hamiltonian ones that can be obtained via this sort of reasoning? (this part might be better asked in a different question if there is an answer that is independent of the context of this one, I'm not sure).

• One sometimes reads that the Legendre transform creates a function that contains all of the information and capability of another function, but in terms of more accessible variables. The price is that the new function has obscure interpretation. It's hard to see this in $L\rightarrow H$. A little easier with the thermodynamic potentials. Easy to see in the analysis of the brushless dc motor where the energy can be written in terms of flux linkage, but how does one measure and control linkage. Much easier to control current. Solution: an energy $\rightarrow$ co-energy Legendre transform. – garyp Apr 14 at 12:13
• Possible duplicate: What's the point of Hamiltonian mechanics? and links therein. – Qmechanic Apr 14 at 12:58
• @Qmechanic I guess I should have linked that one too; I nearly did. But no, I'm not asking what is the point of Hamiltonian, nor Lagrangian, mechanics. I'm asking why specifically the Legendre transform manages to produce a formalism as useful as the Hamiltonian. For example, whether there is some intuition as to why the geometric interpretation of the Legendre transform turns out to be insightful for the description of physical systems. Or whether one can isolate properties of a general transformation between different formalisms that end up pinpointing the Legendre transform. – glS Apr 14 at 13:22
• I just saw that similar questions were raised in the related question physics.stackexchange.com/q/192480/58382. Specifically in some of the comments to the question, and one of the answers. However, the "conclusion" there seems to be that the Legendre transform is the "natural choice" because of its geometrical interpretation. I suppose here I'm inquiring as to whether there is a why to understand why such geometrical features turn out to be insightful for the description of physical systems – glS Apr 14 at 13:28

In Lagrangian/Hamiltonian the crucial property of the Legendre transform is that it preserves all information. One way to demonstrate that the Legendre transform preserves all information is to show that it is its own inverse. When the Legrendre transform is applied twice you are back to the starting point.

The required properties of the transform are very specific, I rather expect that the Legendre tranform is the only transform that has the required properties.

It seems to me that in the end asking what makes the Legendre transformation especially useful is similar to asking what is special about integration by parts that makes it so useful for calculating integrals.

Recommended reading: 2007 article by R. K. P. Zia, Edward F. Redish, and Susan R. McKay
Making sense of the Legendre transform

Blog post by Jess Riedel:
Legendre transform

• the crucial property of the Legendre transform is that it preserves all information. as far as a I can tell, also reading the discussion in physics.stackexchange.com/q/192480/58382, this is far from being true. Applying any invertible transformation to the Lagrangian gives you a different functional which preserves all the information required for the physics. The Legendre transform being an involution might be a more characteristic property though. – glS Apr 14 at 20:43
• It seems to me that in the end asking what makes the Legendre transformation especially useful is similar to asking what is special about integration by parts that makes it so useful for calculating integrals. well, that is also a good question in my opinion, and also a fairly answerable one. See e.g. math.stackexchange.com/a/827604/173147 – glS Apr 14 at 20:45