  This is the first one of two papers on the global dynamics of the original
Boltzmann equations without angular cutoff on the torus. We address the problem
for the hard potentials and Maxwellian molecules in the present paper. The case
of soft potentials is left to a forthcoming paper. The key to solve the problem
is the energy-entropy method which characterizes the propagation of the
regularity, $H$-theorem and the interplay between the energy and the entropy.
Our main results are as follows:
  (i) We present a unified framework to prove the well-posedness for the
original Boltzmann equation for both angular cutoff and without cutoff in
weighted Sobolev spaces with polynomial weights. As a consequence, we obtain an
explicit formula for the asymptotics of the equation from angular cutoff to
non-cutoff.
  (ii) We describe the global dynamics of the equation under the almost optimal
assumption on the solution which makes sure that the Boltzmann collision
operator behaves like a fractional Laplace operator for the velocity variable.
More precisely, we obtain the propagation of the regularity for the solution
and a new mechanism for the convergence of the solution to its equilibrium with
quantitative estimates.
  (iii) We prove that any global and smooth solution to the equation is stable,
i.e., any perturbed solution will remain close to the reference solution if
initially they are close to each other.
