Cubic first integrals of autonomous dynamical systems in $E^2$ by an algorithmic approach

TL;DR

This paper applies a recently developed algorithmic method to compute cubic first integrals of autonomous Newtonian systems in two dimensions, unifying and extending previous results on integrable and superintegrable potentials.

Contribution

It demonstrates the effectiveness of a new theorem-based algorithmic approach to find cubic first integrals, providing an updated reference for integrable potentials in 2D Newtonian systems.

Findings

Collected results in four tables as a reference for integrable potentials.

Extended previous results with new potentials for specific parameter values.

Showed the method's potential to discover new integrable systems with different parameters.

Abstract

In a recent paper (A. Mitsopoulos and M. Tsamparlis, J. Geom. Phys. 170, 104383, 2021), a general theorem is given which provides an algorithmic method for the computation of first integrals (FIs) of autonomous dynamical systems in terms of the symmetries of the kinetic metric defined by the dynamical equations of the system. In the present work, we apply this theorem to compute the cubic FIs of autonomous conservative Newtonian dynamical systems with two degrees of freedom. We show that the known results on this topic, which have been obtained by means of various different methods, and additional ones derived in this work can be obtained by the single algorithmic method provided by this theorem. The results are collected in four Tables which can be used as an updated reference of this type of integrable and superintegrable potentials. The results we find are for special values of free parameters; therefore, using the methods developed here, other researchers by different suitable choice of the parameters will be able to find new integrable and superintegrable potentials.