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.

该研究通过应用机器学习技术（FJet）获得了一个动力学模型，然后使用 Lie 对称技术对这个动力学模型进行分析，得到了运动常数。该分析针对一维和二维谐振子的受守恒与非守恒情况进行，找到了在欠阻尼、过阻尼和临界阻尼情况下的常数。此外，该研究还在各向同性和非各向同性情况下找到了常数，并推广到任意维度。研究还确认了一个常数，使角动量在所有频率之间的比例上得到推广。该研究方法可以从单个通用数据集产生多个运动常数。