It is well known that an almost paracontact manifold is an analogue of the almost contact manifold and it is closely related to almost complex manifold. An almost contact manifold is always odd dimensional on the other hand an almost paracontact manifold could be even dimensional as well. Slant curves are characterized through the scalar product between the normal at the curve and the vertical vector field. The classic mechanic was built on vectors. A formulation of classical mechanics is Lagrangian and Hamiltonian mechanics. The purpose of this paper is the to study Lagrangian and Hamiltonian formalism for mechanical systems using slant curves in 3-dimensional normal almost paracontact metric manifolds, which speaks for a quirky multidisciplinary fields of research. Furthermore, in this study geometricals and physical results concerning to on slant curves in 3-dimensional normal almost paracontact metric manifolds are given.