DSR relativistic particle in a Lagrangian formulation and non-commutative spacetime: A gauge independent analysis

In this Letter we have constructed a coordinate space (or geometric) Lagrangian for a point particle that satisfies the Doubly Special Relativity (DSR) dispersion relation in the Magueijo-Smolin framework. At the same time, the symplectic structure induces a non-commutative phase space, which interpolates between κ-Minkowski and Snyder phase spaces. Hence this model bridges an existing 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 external physical interactions are also briefly outlined. The work serves as a demonstration of how Hamiltonian (and Lagrangian) dynamics can be built around a given non-trivial symplectic structure.