From Poincare Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential

TL;DR

This paper compares Lagrangian descriptors and Poincare maps in analyzing phase space structures of a two-degree-of-freedom Hamiltonian system with a valley ridge inflection point potential, highlighting their effectiveness in understanding reaction dynamics.

Contribution

It introduces a comparative analysis of Lagrangian descriptors and Poincare maps for complex potential energy surfaces with VRI points, a novel application in this context.

Findings

Lagrangian descriptors effectively reveal phase space structures.

Poincare maps provide complementary insights into dynamical pathways.

Both methods help understand trajectory behavior near saddle points.

Abstract

In this paper we compare the method of Lagrangian descriptors with the classical method of Poincare 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 Poincare maps are compared for their ability to determine the phase space structures that govern this dynamical process.