  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.
