  In this paper, we present the results of our investigation relating particle
dynamics and non-commutativity of space-time by using Dirac's constraint
analysis. In this study, we re-parameterise the time $t=t(\tau)$ along with
$x=x(\tau)$ and treat both as configuration space variables. Here, $\tau$ is a
monotonic increasing parameter and the system evolves with this parameter.
After constraint analysis, we find the deformed Dirac brackets similar to the
$\kappa$-deformed space-time and also, get the deformed Hamilton's equations of
motion. Moreover, we study the effect of non-commutativity on the generators of
Galilean group and Poincare group and find undeformed form of the algebra.
Also, we work on the extended space analysis in the Lagrangian formalism. We
find the primary as well as the secondary constraints. Strikingly on
calculating the Dirac brackets among the phase space variables, we obtain the
classical version of $\kappa$-Minkowski algebra.
