Higher order first integrals of autonomous non-Riemannian dynamical systems

TL;DR

This paper develops a systematic method to find first integrals of autonomous non-Riemannian dynamical systems using geometric symmetries, applicable to systems with various orders of integrals.

Contribution

It introduces a theorem that determines first integrals of any order for these systems based on their geometric symmetries, including non-metrical connections.

Findings

The theorem applies to both Riemannian and non-Riemannian systems.

It enables computation of linear, quadratic, and cubic first integrals.

Demonstrated effectiveness on various dynamical systems.

Abstract

We consider autonomous holonomic dynamical systems defined by equations of the form $\ddot{q}^{a}=-\Gamma_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c}$ $-Q^{a}(q)$, where $\Gamma^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a theorem which for these systems determines autonomous and time-dependent first integrals (FIs) of any order in a systematic way, using the `symmetries' of the geometry defined by the dynamical equations. We demonstrate the application of the theorem to compute linear, quadratic, and cubic FIs of various Riemannian and non-Riemannian dynamical systems.