Higher order first integrals of autonomous non-Riemannian dynamical systems

Abstract: We consider autonomous holonomic dynamical systems defined by equations of the form $$\ddot{q}^{{a}=-\Gamma_{bc}^{a}(q)} \dot{q}<sup>{b}\dot{q}<sup>{c}$$ $-Q<sup>{a}(q)$, where $\Gamma<sup>{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q<sup>{a}(q)$ 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.