Lagrangian Descriptors and the Action Integral of Classical Mechanics

TL;DR

This paper brings together the method of Lagrangian descriptors and the principle of least action, or more precisely, of stationary action, in both deterministic and stochastic settings, and shows how the action can be used as a Lagrangia descriptor.

Abstract

In this paper we bring together the method of Lagrangian descriptors and the principle of least action, or more precisely, of stationary action, in both deterministic and stochastic settings. In particular, we show how the action can be used as a Lagrangian descriptor. This provides a direct connection between Lagrangian descriptors and Hamiltonian mechanics, and we illustrate this connection with benchmark examples.

Figures (8)

• Figure 1: A) Total LD with $\tau = 6$ for the linear saddle Hamiltonian system in Eq. \ref{['ds_saddle']} using $\lambda = 1$. B) Backward LD for $\tau = 6$. For the computation of the LD scalar field, all the trajectories starting at the initial conditions on the grid have been integrated for the same time $\tau = 6$. C) The same as B) but the color scale has been adjusted to highlight the unstable manifold.
• Figure 2: A) Total LD with $\tau = 8$ for the linear saddle Hamiltonian system in Eq. \ref{['ds_saddle']} using $\lambda = 1$. B) Stable (blue) and unstable (red) manifolds extracted from the LD scalar field displayed on the left panel. Trajectories have been integrated both forward and backward in time for $\tau = 8$ or until they leave a square $[-16,16]\times[-16,16]$ centered about the origin. This approach is known in the literature as variable time LD.
• Figure 3: Convergence of the long-time average of the forward LD, $\langle \mathcal{S}^{(f)}\rangle$, given in Eq. \ref{['func_conv']}.
• Figure 4: A) and B) Potential energy surface landscape and energy contours for the system parameters $\mathcal{V}^{\ddagger} = 1/4$, $y_w = \sqrt{2}/2$, $\omega = 1$ and $c = 1/2$. We have marked the location of the index-1 saddle at the origin with a red diamond, and the potential wells with red circles. C) Symmetric quartic bistable potential in the $y$ DoF showing the location of the isomerization wells and the transition state. The parameter $\mathcal{V}^{\ddagger}$ measures the potential barrier height measured from the bottom of any of the potential wells, and $y_w$ is the horizontal distance from the index-1 saddle to any of the wells.
• Figure 5: Phase space of the Hamiltonian system at energy $\mathcal{H} = 0.1$, as revealed by Lagrangian descriptors with $\tau = 10$ on the section $\Sigma_1$ in Eq. \ref{['sos']}. A) Backward component of LDs; B) Forward component of LDs; C) Total LD (sum of forward plus backward components); D) Poincaré section superimposed with the stable (blue) and unstable (red) manifolds extracted from the LD scalar field presented in C); E) LD values along the line $p_y = -0.2$ depicted in panel C) to show how the LD scalar field displays 'singular features' at the manifold locations.