Deformed special relativity and deformed symmetries in a canonical framework

In this paper we have studied the nature of kinematical and dynamical laws in κ-Minkowski spacetime from a new perspective: the canonical phase space approach. We discuss a particular form of κ-Minkowski phase space algebra that yields the κ-extended finite Lorentz transformations derived in [D. Kimberly, J. Magueijo, and J. Medeiros, Phys. Rev. D 70, 084007 (2004).]. This is a particular form of a deformed special relativity model that admits a modified energy-momentum dispersion law as well as noncommutative κ-Minkowski phase space. We show that this system can be completely mapped to a set of phase space variables that obey canonical (and not κ-Minkowski) phase space algebra and special relativity Lorentz transformation (and not κ-extended Lorentz transformation). The complete set of deformed symmetry generators are constructed that obeys an unmodified closed algebra but induce deformations in the symmetry transformations of the physical κ-Minkowski phase space variables. Furthermore, we demonstrate the usefulness and simplicity of this approach through a number of phenomenological applications both in classical and quantum mechanics. We also construct a Lagrangian for the κ-particle.