First integrals of holonomic systems without Noether symmetries

TL;DR

This paper presents a theorem linking first integrals of holonomic systems to collineations of the kinetic metric, enabling the discovery of new integrals beyond standard methods through an algorithmic approach.

Contribution

It introduces a novel theorem connecting first integrals to kinetic metric collineations and demonstrates an algorithm to find new integrals not accessible by traditional techniques.

Findings

The theorem relates first integrals to collineations of the kinetic metric.

The approach can find previously unknown first integrals.

Examples show the method's effectiveness in discovering new integrals.

Abstract

A theorem is proved which determines the first integrals of the form $I=K_{ab}(t,q)\dot{q}^{a}\dot{q}^{b}+K_{a}(t,q)\dot{q}^{a}+K(t,q)$ of autonomous holonomic systems using only the collineations of the kinetic metric which is defined by the kinetic energy or the Lagrangian of the system. It is shown how these first integrals can be associated via the inverse Noether theorem to a gauged weak Noether symmetry which admits the given first integral as a Noether integral. It is shown also that the associated Noether symmetry is possible to satisfy the conditions for a Hojman or a form-invariance symmetry therefore the so-called non-Noetherian first integrals are gauged weak Noether integrals. The application of the theorem requires a certain algorithm due to the complexity of the special conditions involved. We demonstrate this algorithm by a number of solved examples. We choose examples from published works in order to show that our approach produces new first integrals not found before with the standard methods.