Doubly Special Relativity and κ-Minkowski Non-Commutative Spacetime: Towards a Dynamical Connection

In this paper we have constructed a coordinate space (or geometric) Lagrangian for a point particle that satisfies the Doubly Special Relativity dispersion relation in the Magueijo-Smolin framework. At the same time, the symplectic structure induces a Non-Commutative κ-Minkowski spacetme. Hence this model bridges a gap between two conceptually distinct ideas in a natural way. We thoroughly discuss how this type of construction can be carried out from a phase space (or first order) Lagrangian approach. The inclusion of interactions are briefly outlined. The work serves as a demonstration of how Hamiltonian (and Lagrangian) dynamics can be built around a given non-trivial symplectic structure.