In the present talk we will discuss two different problems both of which are related (indirectly and directly) to Cosmology. In (I) we will develop the dynamical model of a point particle in a specific NonCommutative spacetime, that conforms to Doubly Special Relativity (DSR) dispersion relation. In (II) we will describe our formulation of Chaplygin gas model in Cosmology and our result tends to disagree with the conventional scenario. (I): Theoretical results in High Energy Physics and Cosmology suggests the possibility of a discrete spacetime below Planck length, whereas some astrophysical observations seem to require a modification in the particle energymomentum relation, that agrees with DSR. Interestingly both of them induce a NonCommutative spacetime. We have recently constructed a dynamical (Hamiltonian and Lagrangian) model of a point particle that lives in a NonCommutative spacetime and obeys a particular form of DSR energymomentum relation. (II): The interest in Chaplygin gas model in Cosmology stems from the fact that in this model, the evolution of the Universe from a dust dominated phase to the present day Cosmological Constant phase (that leads to the accelerated expansion of the Universe), is described by a single fluid. However, we feel that the conventional derivation yielding that result is not totally correct. We consider a fully relativistic Action formulation and find that the Chaplygin fluid can give rise to only a dust dominated phase.
