Dynamical Systems induced by Canonical Divergence in dually flat manifolds

Abstract: 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.