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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."

Key Insights Distilled From

by Michael F. Z... at **arxiv.org** 03-29-2024

Deeper Inquiries

To extend the approach to more complex dynamical systems beyond the harmonic oscillator, one could consider systems with higher degrees of freedom, such as multi-particle systems or systems with non-linear interactions. The key would be to generalize the Lie symmetry techniques and the FJet method to accommodate the increased complexity. This may involve developing new algorithms or mathematical frameworks to handle the additional variables and interactions present in these systems. Additionally, incorporating more sophisticated machine learning models that can capture the intricacies of these systems would be essential for accurate modeling and analysis.

One limitation of the Lie symmetry analysis in deriving constants of motion is that it may not always be straightforward to identify the appropriate symmetries for complex systems. In such cases, the analysis may become computationally intensive and challenging. To overcome this limitation, one could explore advanced computational techniques, such as numerical methods or symbolic computation, to assist in identifying the symmetries and deriving the constants of motion. Additionally, collaborating with experts in the specific field of study to gain insights into the underlying physics of the system could provide valuable guidance in the analysis process.

The insights gained from the work on constants of motion can be applied to improve the interpretability and robustness of machine learning models for physical systems in several ways. Firstly, by incorporating the derived constants of motion as additional constraints or features in the machine learning models, the models can be guided to adhere to the fundamental principles governing the dynamics of the system. This can enhance the interpretability of the models by ensuring that they align with known physical laws. Secondly, by utilizing the constants of motion to validate and calibrate the machine learning models, their robustness and accuracy can be improved, leading to more reliable predictions and analyses of physical systems. Additionally, the integration of domain knowledge, such as the conservation laws represented by the constants of motion, can help in building more explainable and trustworthy machine learning models for complex physical systems.

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