Accelerating Cosmological Model with Scalar Field in $f(R,\mathcal{L}_{m})$ Gravity

In this work, we investigate the cosmological dynamics of $f(R,\mathcal{L}_m)$ gravity using two complementary scenarios: without a scalar field and with a minimally coupled generalized scalar field. For the case without a scalar field, we consider a functional form with linear and exponential dependence on the matter Lagrangian and perform a dynamical system analysis. The resulting autonomous system admits a matter-dominated saddle configuration and a de Sitter attractor . Due to the non-hyperbolic nature of the critical curves, Center Manifold Theory (CMT) is employed to establish the local asymptotic stability of the de Sitter solution. We then extend the framework by including a minimally coupled generalized scalar field with an exponential self-interacting potential. The extended autonomous system also contains a matter-dominated saddle point and a stable dark-energy-dominated attractor corresponding to a late-time de Sitter phase. The stability of the attractor is confirmed through CMT, and the evolution of the cosmological parameters demonstrates a smooth transition from a matter-dominated decelerating era to accelerated. Also, in the absence of the scalar field, the matter-dominated configuration associated with a vanishing nonlinear contribution remains stable, whereas the inclusion of the scalar field transforms the matter era into a saddle configuration, thereby enabling the Universe to naturally evolve toward the late-time accelerated attractor. In both scenarios, the exponential power term equal to $-1$ corresponds to a late-time de Sitter phase. These results show that both scenarios lead to late-time cosmic acceleration, while the inclusion of the scalar field provides an additional dynamical mechanism for realizing a stable dark-energy-dominated Universe without invoking a cosmological constant.