We calculate the Hamiltonian as the Legendre transform of the Lagrangian $$H(q,p,t) = p \dot q - L(q,\dot q,t), $$ where $p$ is the slope function $$p \equiv \frac{\partial L}{\partial \dot q} .$$
The general idea of the Legendre transform is that we replace our function $f(x)$, depending on some variable $x$, with a new function $F(s)$ that only depends on the slope $$s(x) = \frac{\partial f(x)}{\partial x} $$ which, however, contains the same information.
For a function which depends on multiple variables $g(x,y,z,\ldots)$ we can do this for any of the variables and generate this way different Legendre transforms.
Therefore, I was wondering if it also makes sense to consider the alternative Legendre transform of the Lagrangian $$G(s,\dot q,t) = s q - L(q,\dot q,t), $$ where $s$ is the slope function $$s \equiv \frac{\partial L}{\partial q} .$$ If yes, what's the name of this alternative Legendre transform? And if not, why does it make sense to consider the Legendre transform with respect to $\dot q$ but not with respect to $q$?
PS: I found a related idea in this paper but they consider Legendre transforms involving $ \dot p \equiv \frac{\partial H}{\partial q} .$
