Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, and their stable and unstable manifolds in molecular Hamiltonian have been an integral part of computing quantitative results such as the cumulative reaction probability and rate constants in chemical reactions. Thus, methods that can reveal the geometry of these invariant manifolds in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In these articles, we assessed the capability of one such method called Lagrangian descriptor (LD) for revealing the aforementioned high dimensional phase space structures associated with an index-1 saddle in Hamiltonian systems. The LD 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 LD contour maps and the phase space structures’ intersection with an isoenergetic two-dimensional surface, and hence provides a verification of the method. Next, the method of LD is applied to classical two and three degrees of freedom Hamiltonians that model features of dissociation reactions. The result of the LD based approach is compared with an established numerical method for computing unstable periodic orbit and tube manifolds of a two degrees of freedom system, and the results are in good agreement. We have also discussed the results in the context of three degrees of freedom extension of the same model Hamiltonian.
Shibabrat Naik, Víctor J. García-Garrido, Stephen Wiggins, Finding NHIM: Identifying high dimensional phase space structures in reaction dynamics using Lagrangian descriptors, Communications in Nonlinear Science and Numerical Simulation, 2019, 79, 104907
Shibabrat Naik and Stephen Wiggins, Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential, Phys. Rev. E, 2019, 100 (2), 022204