Constants of Motion for Conserved and Non-conserved Dynamics

TL;DR

This work combines the FJet machine-learning framework with Lie symmetry analysis to extract constants of motion from time-series data for both conserved and dissipative dynamics. By deriving integrating factors and explicit invariants for 1D and 2D harmonic oscillators (including underdamped, overdamped, and critically damped regimes, as well as anisotropic/isotropic cases), the authors show how a single data set can yield multiple independent constants. A key interpretation is that these constants reflect the conservation of the total energy of the oscillator plus dissipative environment, and the approach generalizes to arbitrary dimensions, yielding angular-momentum-like invariants. The methodology promises robust domain knowledge extraction from minimal data and connects symmetry principles with data-driven modeling for broad physical systems.

Abstract

This paper begins with a dynamical model that was obtained by applying a machine learning technique (FJet) to time-series data; this dynamical model is then analyzed with Lie symmetry techniques to obtain constants of motion. This analysis is performed on both the conserved and non-conserved cases of the 1D and 2D harmonic oscillators. For the 1D oscillator, constants are found in the cases where the system is underdamped, overdamped, and critically damped. The novel existence of such a constant for a non-conserved model is interpreted as a manifestation of the conservation of energy of the {\em total} system (i.e., oscillator plus dissipative environment). For the 2D oscillator, constants are found for the isotropic and anisotropic cases, including when the frequencies are incommensurate; it is also generalized to arbitrary dimensions. In addition, a constant is identified which generalizes angular momentum for all ratios of the frequencies. The approach presented here can produce {\em multiple} constants of motion from a {\em single}, generic data set.

Figures (4)

• Figure 1: This figure displays the first two sheets ($n=0,1$) for $r'(u,v;n) = e^r$, using $r$ from Eq. \ref{['eqn:r_under_main']}; each is a combination contour/heatmap plot. The parameters used were $\omega_0 =1$ and $\gamma = 0.1$ (i.e., the underdamped case). The horizontal green line indicates a branch cut. The reddish-colored curves represent level sets of $r'$, and the blue dots represent a sampling of the trajectory computed using the exact solution. Note the blue dots stay on a single value of $r'$, as they should. The shading and contour values are consistent between the two plots.
• Figure 2: In this figure for the overdamped case (with $\omega_0 =1$, $\gamma=1.1$), a contour plot of $r$ (from Eq. \ref{['eqn:r_over_main']}) is shown on the left, and a heatmap on the right. On the left plot, blue dots signify a sampled trajectory at time steps of $0.2$. Note the blue dots stay on a single value of $r$, as they should. In the right plot, the heatmap reveals the location of the positive/negative singularities along the two lines. For graphing purposes, the minimum value allowed for $|\zeta u \pm (\gamma u + v) |$ was $0.001$.
• Figure 3: In this figure for the critically damped case (with $\omega_0 = \gamma=1$), a contour plot of $r$ (from Eq. \ref{['eqn:r_critical_main']}) is shown on the left, and a heatmap on the right. On the left plot, blue dots signify a sampled trajectory at time steps of 0.2. Note the blue dots stay on a single value of $r$, as they should. In the right plot, the heatmap reveals the location of the positive/negative singularities where $| \gamma u + v|$ equaled zero; its minimum value was limited to $0.01$ for graphing purposes.
• Figure 4: For the overdamped case, the $u,v$-plane is divided up into four regions, according to the signs of $(\tilde{u},\tilde{v})$. In the left plot, for example, the region marked $+-$ corresponds to where $\tilde{u}>0$ and $\tilde{v}<0$. These four regions are due to divergences in $r$, which occur where $\zeta u \pm (\gamma u + v) = 0$. The right plot displays an example of a starting point at $(\tilde{u}_0,\tilde{v}_0) = (1,1)$ (shown as a red dot), and the other corresponding starting points (as open circles) at $(1,-1)$, $(-1,1)$, and $(-1,-1)$, according to the discussion given.