A goal of CHAMPS is to bring together developments in mathematics with problems in chemistry that reveal a new and unique insight. The method of Lagrangian descriptors is an excellent example of such a synergy.
Lagrangian descriptors are a trajectory diagnostic for revealing phase space structures in dynamical systems. The method was originally developed in the context of Lagrangian transport studies in fluid dynamics (Madrid and Mancho 2009) but the wide applicability of the method has recently been recognized in chemistry, see (Craven and Hernandez 2015, Junginger and Hernandez 2016, Feldmaier, Junginger et al. 2017, Junginger, Duvenbeck et al. 2017, Junginger, Main et al. 2017).
The method is very compelling since it is straightforward to implement computationally, the interpretation is evident, and it provides a “high resolution” method for exploring high dimensional phase space with low dimensional slices. It also applies to both Hamiltonian and non-Hamiltonian systems (Lopesino, Balibrea-Iniesta et al. 2017) as well as to systems with arbitrary, even stochastic, time-dependence (Balibrea-Iniesta, Lopesino et al. 2016). Moreover, Lagrangian descriptors can be applied directly to data sets, without the need of an explicit dynamical system (Mendoza, Mancho et al. 2014).
Briefly, Lagrangian descriptors are implemented as follows. Each point in a chosen grid of initial conditions for trajectories in phase space is assigned a value according to the arclength of the trajectory starting at that initial condition after integration for a fixed, finite time, both backward and forward in time (all initial conditions in the grid are integrated for the same time). The idea is that the influence of phase space structures on trajectories will result in differences in arclength of nearby trajectories near a phase space structure. This has been quantified in terms of the notion of “singular structures’ in the Lagrangian descriptor plots, which are easy to recognize visually (Mancho, Wiggins et al. 2013, Lopesino, Balibrea-Iniesta et al. 2017).
Trajectories are the “primitive objects” that are used to explore phase space structure. In fact, phase space structure is “built” from trajectories. For high dimensional phase space this approach is problematic and prone to issues of interpretation since a tightly grouped set of initial conditions may result in trajectories that become “lost” with respect to each other in phase space. The method of Lagrangian descriptors turns this problem on its head by emphasizing the initial conditions of trajectories, rather than the precise location of their futures and pasts, after a specified amount of time. A low dimensional “slice” of phase space can be selected and sampled with a grid of initial conditions of high resolution. Since the phase space structure is encoded in the initial conditions of the trajectories, no resolution is lost as the trajectories evolve in time.
We remark that the original Lagrangian trajectory diagnostic is the arclength. This has been modified in (Lopesino, Balibrea et al. 2015, Lopesino, Balibrea-Iniesta et al. 2017) where, effectively, a different type of norm is used as a trajectory diagnostic. This has been shown to have many advantages over the arclength. For example, it allows for a rigorous analysis of the notion of “singular structures” in certain cases and the relation of this notion to invariant manifolds. It also allows a natural decomposition of the Lagrangian descriptor in a way that isolates distinct dynamical effects. This was utilized in (Demian and Wiggins 2017) in order to show that a Lagrangian descriptor could be used to detect the Lyapunov periodic orbits in the two degree-of-freedom Henon-Heiles Hamiltonian system.
This figure is from (Mendoza, Mancho et al. 2014) and shows the results of the use of the Lagrangian descriptor for revealing flow structures in the Gulf Stream using a geophysical fluid dynamics data set.
The use and further development of Lagrangian descriptors is a topic underlying many of the themes of CHAMPS: from understanding the role of phase space structure in high dimensional systems to dimensional reduction and as well as for machine learning from data sets.
Balibrea-Iniesta, F., et al. (2016). “Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems.” International Journal of Bifurcation and Chaos 26(13): 1630036.
Craven, G. T. and R. Hernandez (2015). “Lagrangian descriptors of thermalized transition states on time-varying energy surfaces.” Physical review letters 115(14): 148301.
Demian, A. S. and S. Wiggins (2017). “Detection of Periodic Orbits in Hamiltonian Systems Using Lagrangian Descriptors.” International Journal of Bifurcation and Chaos 27(14): 1750225.
Feldmaier, M., et al. (2017). “Obtaining time-dependent multi-dimensional dividing surfaces using Lagrangian descriptors.” Chemical Physics Letters 687: 194-199.
Junginger, A., et al. (2017). “Chemical dynamics between wells across a time-dependent barrier: Self-similarity in the Lagrangian descriptor and reactive basins.” The Journal of chemical physics 147(6): 064101.
Junginger, A. and R. Hernandez (2016). “Lagrangian descriptors in dissipative systems.” Physical Chemistry Chemical Physics 18(44): 30282-30287.
Junginger, A., et al. (2017). “Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points.” Physical Review A 95(3): 032130.
Lopesino, C., et al. (2015). “Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps.” Communications in Nonlinear Science and Numerical Simulation 27(1): 40-51.
Lopesino, C., et al. (2017). “A theoretical framework for lagrangian descriptors.” International Journal of Bifurcation and Chaos 27(01): 1730001.
Madrid, J. J. and A. M. Mancho (2009). “Distinguished trajectories in time dependent vector fields.” Chaos: An Interdisciplinary Journal of Nonlinear Science 19(1): 013111.
Mancho, A. M., et al. (2013). “Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems.” Communications in Nonlinear Science and Numerical Simulation 18(12): 3530-3557.
Mendoza, C., et al. (2014). “Lagrangian descriptors and the assessment of the predictive capacity of oceanic data sets.” Nonlinear Processes in Geophysics 21(3): 677-689.