  In this paper, the expanding nature of the universe within the context of
$f(R, L_m)$ gravity theory, where $R$ is Ricci Scalar and $L_m$ is matter
Lagrangian, is explored. By employing a specific form $f(R, L_m) = \Lambda +
\frac{\alpha}{2} R + \beta L_m^n$, where $\Lambda$, $\alpha$, $\beta$, $n$ are
model parameters, with the matter Lagrangian given by $L_m = -p$. We solve the
field equations by considering the equation of state (EoS) parameter $\omega$,
defined as $p = \omega \rho $, where $p$ represents the isotropic pressure and
$\rho$ represents the energy density of the perfect fluid. We thoroughly
investigate the model through phase plane analysis, actively studying the
evolution of cosmological solutions using dynamical system techniques. To
analyse the evolution equations, we have introduced suitable transformation of
variables. The evolution of corresponding solutions is presented by curves in
the phase-plane diagram. The nature of critical points are analysed and stable
attractors are examined for $f(R, L_m)$ gravity cosmological model. We
determine the classical stabilities of these cosmologies. We also examine the
transition of early decelerating stage of the universe to present accelerating
stage of the universe. We analyze the evolution of the effective equation of
state parameters and statefinder diagnostics for different parameter values. We
examine and analyze the nature of critical points from a cosmological
perspective.
