In standard Mechanics textbooks Hamiltonian is introduced in a very ad hoc way: Let's try these Legendre transformations and see what happens. Wow, the equations turned out to be more symmetric. How Lucky! I am looking for a less serendipitous path from Lagrangian to Hamiltonian.
The most natural way of writing Lagrange equation as a pair of first order equations is:
$\frac{dq}{dt} = \dot{q}$
$\frac{d\dot{q}}{dt} = F(q, \dot{q}, t)$
What's a principled way of coming up with a transformation of variables that will make the set of equations maximally symmetric?
Also, how can I tell by looking at this vector flow that it will turn out to be $\Omega dH$ with simplectic $\Omega$ rather than just $dH$?