This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space–time endowed with a suitable metric due to Eisenhart. Until now, this framework has never been given attention to describe chaotic dynamics. A gap that is filled in the present work. In a Riemannian-geometric context, the stability/instability of the dynamics depends on the curvature properties of the ambient manifold and is investigated by means of the Jacobi–Levi-Civita (JLC) equation for geodesic spread. It is confirmed that the dominant mechanism at the ground of chaotic dynamics is parametric instability due to curvature variations along the geodesics. A comparison is reported of the outcomes of the JLC equation written also for the Jacobi metric on configuration space and for another metric due to Eisenhart on an extended configuration space–time. This has been applied to the Hénon–Heiles model, a two-degrees of freedom system. Then the study has been extended to the 1D classical Heisenberg $XY$ model at a large number of degrees of freedom. Both the advantages and drawbacks of this geometrization of Hamiltonian dynamics are discussed. Finally, a quick hint is put forward concerning the possible extension of the differential–geometric investigation of chaos in generic dynamical systems, including dissipative ones, by resorting to Finsler manifolds.

本文利用黎曼几何的基本工具解决了哈密顿的混沌问题。更准确地说，将哈密顿流识别为配置空间上的测地线流-由于艾森哈特（Eisenhart）而赋予了适当的度量。到现在为止，这个框架从未被用来描述混沌动力学。当前工作填补了空白。在黎曼几何环境中，动力学的稳定性/不稳定性取决于环境歧管的曲率特性，并通过雅可比-列维-奇维塔（JLC）方程进行了测地线传播的研究。可以确认，由于沿测地线的曲率变化，混沌动力学基础上的主导机制是参数不稳定性。有报告比较了JLC方程的结果，该方程还为配置空间上的Jacobi度量以及在扩展的配置时空上由于Eisenhart导致的另一个度量写了。这已应用于Hénon-Heiles模型（一个两自由度系统）。然后将研究扩展到一维经典海森堡$Xÿ$大量自由度的模型。讨论了汉密尔顿动力学几何化的优点和缺点。最后，通过使用Finsler流形，对有关一般动力学系统（包括耗散系统）中混沌的微分几何研究的可能扩展提出了一个简短的提示。