Higher order first integrals of autonomous dynamical systems

TL;DR

This paper presents a theorem for finding higher order first integrals in autonomous dynamical systems using collineations and Killing tensors, and applies it to identify integrable potentials in Newtonian systems.

Contribution

It introduces a general theorem linking first integrals to geometric symmetries and applies it to discover new integrable potentials with cubic first integrals.

Findings

Identified known and new superintegrable potentials with cubic first integrals.

Developed a method to compute higher order first integrals using geometric properties.

Extended the classification of integrable systems in Newtonian mechanics.

Abstract

A theorem is derived which determines higher order first integrals of autonomous holonomic dynamical systems in a general space, provided the collineations and the Killing tensors -- up to the order of the first integral -- of the kinetic metric, defined by the kinetic energy of the system, can be computed. The theorem is applied in the case of Newtonian autonomous conservative dynamical systems of two degrees of freedom, where known and new integrable and superintegrable potentials that admit cubic first integrals are determined.