Finslerian Lagrangian In A New Non-Commutative Space Sudipta Das a, 1 , Subir Ghosh a, 2 and Salvatore Mignemi b, 3 a Physics and Applied Mathematics Unit, Indian Statistical Institute 203 B. T. Road, Kolkata 700108, India b Dipartimento di Matematica, Universita’ di Cagliari viale Merello 92, 09123 Cagliari, Italy and INFN, Sezione di Cagliari Abstract We propose a new non-canonical phase space with operatorial form of noncommutativity. The Lorentz generators induce deformed transformations. However, the Lorentz algebra itself is deformed only at a higher order of Non-Commutative (NC) parameters. A dual canonical phase space (through the Darboux map) is provided which is instrumental in constructing a point particle Lagrangian residing in this non-canonical phase space. The overall consistency of the scheme and results is achieved by identifying the NC algebra as Dirac Brackets. The resulting Lagrangian is of Finsler form. Fax (Subir Ghosh): +91(33) 25773026 Telephone (Subir Ghosh): +91(33) 25753362 1 E-mail: sudipta.das r@isical.ac.in 2 E-mail: sghosh@isical.ac.in 3 E-mail: smignemi@unica.it 1