标题： 雅可比度规与动力系统的莫尔斯理论 显示英文标题

标题： Jacobi Metric and Morse Theory of Dynamical Systems

摘要： 将 Maupertuis 原理推广到二阶变分法中已完成。 因此，通过扩展 Jacobi 定理来分析自然动力系统的解的稳定性。 结果表明，尽管作用量和 Jacobi 长度周围的二阶变分泛函不一致，但动力系统中轨迹的 Morse 理论与 Jacobi 度量中测地线的 Morse 理论是相同的。 作为一个代表性例子，我们将此结果应用于研究 Garnier 系统的分隔解。 显示英文摘要

摘要： The generalization of the Maupertuis principle to second-order Variational Calculus is performed. The stability of the solutions of a natural dynamical system is thus analyzed via the extension of the Theorem of Jacobi. It is shown that the Morse Theory of the trajectories in the dynamical system is identical to the Morse Theory of geodesics in the Jacobi metric, even though the second-variation functionals around the action and the Jacobi length do not coincide. As a representative example, we apply this result to the study of the separatrix solutions of the Garnier System.