Core Concepts
The existence of constants of motion in both conserved and non-conserved dynamical systems can be discovered by combining machine learning techniques (FJet) to model the dynamics with Lie symmetry analysis.
Abstract
The paper begins by introducing a dynamical model obtained using the FJet machine learning technique on time-series data. It then analyzes this model using Lie symmetry techniques to derive constants of motion for both conserved and non-conserved cases of the 1D and 2D harmonic oscillators.
For the 1D oscillator, constants are found for the underdamped, overdamped, and critically damped cases. The existence of a constant for the non-conserved model is interpreted as a manifestation of the conservation of energy of the total system (oscillator plus dissipative environment).
For the 2D oscillator, constants are found for the isotropic and anisotropic cases, including when the frequencies are incommensurate. A constant is also identified which generalizes angular momentum for all ratios of the frequencies.
The approach presented can produce multiple constants of motion from a single, generic data set, demonstrating the power of combining machine learning and analytical techniques.
Stats
The paper does not contain any explicit numerical data or statistics. The focus is on deriving analytical expressions for the constants of motion.
Quotes
"The novel existence of such a constant for a non-conserved model is interpreted as a manifestation of the conservation of energy of the total system (i.e., oscillator plus dissipative environment)."
"A constant is identified which generalizes angular momentum for all ratios of the frequencies."
"The approach presented here can produce multiple constants of motion from a single, generic data set."