  The principles of classical mechanics have shown that the inertial quality of
mass is characterized by the kinetic energy. This, in turn, establishes the
connection between geometry and mechanics. We aim to exploit such a fundamental
principle for information geometry entering the realm of mechanics. According
to the modification of curve energy stated by Amari and Nagaoka for a smooth
manifold $\mathrm{M}$ endowed with a dual structure
$(\mathrm{g},\nabla,\nabla^*)$, we consider $\nabla$ and $\nabla^*$ kinetic
energies. Then, we prove that a recently introduced canonical divergence and
its dual function coincide with Hamilton principal functions associated with
suitable Lagrangian functions when $(\mathrm{M},\mathrm{g},\nabla,\nabla^*)$ is
dually flat. Corresponding dynamical systems are studied and the tangent
dynamics is outlined in terms of the Riemannian gradient of the canonical
divergence. Solutions of such dynamics are proved to be $\nabla$ and $\nabla^*$
geodesics connecting any two points sufficiently close to each other.
Application to the standard Gaussian model is also investigated.
