In this paper the mathematical idealization of the classical variational problem in Lorentz-Minkowski space is studied for the curve α which is timelike, spacelike and lightlike, parameterized by the pseudo-length-arc. The geodesic curvature and torsion of an elastic curve are evaluated if they exist as the solutions of the differential equations for all different cases. Due to elastic curve definition, the minimum principle and the Noethers’ Theorem are applied to elastic energy function which is defined as the integral of the squared second derivative norm of the curve.