The general Randers spaces were introduced by R. Miron [2]. These are some generalizations of Randers spaces denoted GRⁿ = (M; F + β), equipped with the Lorentz nonlinear connection. In the present paper we dene the General Randers Mechanical System as a triple (M; T; F_e), where T is the energy of the Rⁿ space. We obtain the expressions for the curvature and the torsion of GRⁿ and we give the formula for the local coecients of the canonical connection.