Huygens' equations and the gradient-flow equations in information geometry

We revisited the relation between the gradient equations and Hamilton's equations in information geometry. By regarding a gradient-flow equation in information geometry as Huygens' equation in geometric optics, we relate the gradient flow to the geodesic flow induced by a geodesic Hamiltonian in Riemannian geometry. The original time parameter $t$ in the gradient equations is related to the arc-length parameter in the Riemannian manifold by Jacobi-Maupertuis transformation. As a by-product, it is found the relation between the gradient equation and replicator equations.