Higher order first integrals of autonomous non-Riemannian dynamical systems

We consider autonomous holonomic dynamical systems defined by equations of the form ${\ddot{q}}^a=-{\mathrm{\Gamma }}_{bc}^a(q){\dot{q}}^b{\dot{q}}^c$ $-Q^a(q)$, where ${\mathrm{\Gamma }}_{bc}^a(q)$ 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.