# Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors

## TL;DR

This paper evaluates the effectiveness of Lagrangian descriptors in identifying high-dimensional phase space structures, such as NHIMs, in Hamiltonian systems, by comparing computational results with known analytical solutions.

## Contribution

The study demonstrates that Lagrangian descriptors can accurately reveal high-dimensional phase space structures in Hamiltonian systems, validated against known analytical solutions.

## Key findings

- Lagrangian descriptors successfully identify phase space structures.
- The method provides clear visualization of high-dimensional structures.
- Validation against analytical solutions confirms accuracy.

## Abstract

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures' intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.