Constants of Motion for Conserved and Non-conserved Dynamics
Michael F. Zimmer
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.