In this paper we compare the method of Lagrangian descriptors with the classical method of Poincaré maps for revealing the phase space structure of two-degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a two-degree-of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincaré maps are compared for their ability to determine the phase space structures that govern this dynamical process.

在本文中，我们将拉格朗日描述符的方法与庞加莱图的经典方法进行了比较，以揭示两自由度哈密顿系统的相空间结构。比较是通过考虑具有谷脊拐点（VRI）势能面的两自由度系统的动力学来进行的。VRI势能面有四个关键点：将两个井分开的高能鞍座和低能鞍座。在两个鞍形点之间是一个谷脊弯曲点，这是势能表面几何形状从谷向脊变化的点。两个鞍座之间的区域形成一个反应通道，感兴趣的动力学问题是轨迹如何穿越高能鞍座，向低能鞍座发展，并选择要输入的特定孔。将拉格朗日描述符和庞加莱图进行比较，以确定它们确定控制此动态过程的相空间结构的能力。