My original interest, for asking this question, is to better understand the use of the Legendre transform in the Gibbons-Hawking ansatz, or its higher-dimensional generalization by Hitchin-Karlhede-Lindstrom-Rocek. While I do understand the mathematical derivation, my question is more of the type: a priori, is there a method to justify the use of a Legendre transform? Can one know before-hand that using such a coordinate transformation (or some other transformation), will lead to some simpler differential equations?
In order to avoid a discussion-type post, I will try to narrow it down. I will be happy with a justification in the more elementary setting of classical mechanics: going from the Euler-Lagrange equations to the Hamilton equations, for say the motion of a single particle.
My question is this. Question 1. Starting from the Euler-Lagrange equations for a single particle, is there a machinery available, that will tell us that using a Legendre transform will produce simpler differential equations (namely the Hamilton equations). Here I view the Legendre transform as a change of coordinates in the first order jet manifold $J^1(M)$, where $M$ is $\mathbb{R}^4$ with independent coordinate $t$ and dependent coordinates $q^1, q^2, q^3$, and $\dot{q}^1, \dot{q}^2, \dot{q}^3$, denote the extra first order coordinates in $J^1(M)$.
I have the impression that the people working with Exterior Differential Systems and Cartan's methods might know of a justification perhaps. Kindly share!
Here is an extra question which is in the same spirit. Question 2. Let $M = \mathbb{R}^{m+1}$, with independent variables $x^1$,...,$x^m$, and dependent variable $u$. If $F: J^n(M) \to \mathbb{R}$ is a differential function, where $J^n(M)$ is the $n$-th order jet space of $M$, then $F = 0$ defines a differential equation of order $n$ for a function of $m$ variables. My question is now this. How much can one straighten the hypersurface $F=0$ in $J^n(M)$, while preserving the contact forms? (This basically amounts to asking, how much can one simplify the form of the differential equation $F=0$, using a local change of coordinate in $J^n(M)$?)