  We revisit the relation between the gradient-flow equations and Hamilton's
equations in information geometry. By regarding the gradient-flow equations as
Huygens' equations in geometric optics, we have related the gradient flows to
the geodesic flows induced by the geodesic Hamiltonian in an appropriate
Riemannian geometry. The original evolution parameter $\textit{t}$ in the
gradient-flow equations is related to the arc-length parameter in the
associated Riemannian manifold by Jacobi-Maupertuis transformation. As a
by-product, it is found the relation between the gradient-flow equation and
replicator equations.
