Noether symmetry and exact solutions in f(Q,ϕ) gravity with scalar field dynamics

This study explores the application of the Noether symmetry approach within the framework of extended symmetric teleparallel [Formula: see text] gravity, which integrates the non-metricity scalar [Formula: see text] and a scalar field [Formula: see text], in the context of a Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. By employing this approach, we investigate a specific form of [Formula: see text] and examine the existence of a nontrivial Noether vector. The Noether symmetry method proves instrumental in constructing models and deriving exact solutions for the Lagrangian. We demonstrate how this method determines the functional form of [Formula: see text] and yields precise cosmological solutions. These solutions align with the symmetries identified and suggest an accelerated expansion of the universe. Furthermore, we compute key cosmological parameters, including the Hubble parameter, deceleration parameter, and effective equation of state parameter, which are consistent with current observational data.