# Stellar Dynamical Modeling -- Counting Conserved Quantities

## TL;DR

This study compares two methods for counting conserved quantities in stellar orbits within galaxies, highlighting their limitations and performance, and discusses potential for future trajectory analysis tools.

## Contribution

It evaluates correlation integral and machine learning methods for identifying conserved quantities from stellar orbit trajectories, revealing their current shortcomings and computational performance.

## Key findings

- Neither method fully recovers the number of conserved quantities.
- Correlation integral approach is faster computationally.
- Determining explicit algebraic formulas for conserved quantities remains challenging.

## Abstract

Knowing the conserved quantities that a galaxy's stellar orbits conform to is important in helping us understand the stellar distribution and structures within the galaxy. Isolating integrals of motion and resonances are particularly important, non-isolating integrals less so. We compare the behavior and results of two methods for counting the number of conserved quantities, one based on the correlation integral approach and the other a more recent method using machine learning. Both methods use stellar orbit trajectories in phase space as their only input, and we create such trajectories from theoretical spherical, axisymmetric and triaxial model galaxies. The orbits have known isolating integrals and resonances. We find that neither method is fully effective in recovering the numbers of these quantities, nor in determining the number of non-isolating integrals. From a computer performance perspective, we find the correlation integral approach to be the faster. Determining the algebraic formulae of (multiple) conserved quantities from the trajectories has not been possible due to the lack of an appropriate symbolic regression capability. Notwithstanding the shortcomings we have noted, it may be that the methods are usable as part of a trajectory analysis tool kit.