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.

翻译：我们在信息几何中重新审视了梯度方程与汉密尔顿方程之间的关系。关于信息几何中的梯度-流量方程作为Huygens的几何光学等式,我们将梯度流与里曼尼几何学中一位大地测量学家汉密尔顿所引的大地测量流联系起来。梯度方程中的原时间参数$t美元与Jacobi-Maupertuis转换的里曼方程中的弧长参数有关。作为一个副产品,我们发现了梯度方程与翻版方程之间的关系。