Any formulation of equations of motion requires characterization of the role of the physical restrictions that are imposed on a system's movement. These restrictions lead to kinematical relations between motion variables, and they also are manifested as reaction forces. When a system consists of interconnected bodies, the standard Newton–Euler formulation isolates individual bodies. The need to account for the kinematical constraints and corresponding reaction forces associated with each connection substantially enhances the level of effort entailed in deriving equations of motion.

The Lagrangian formulation implicitly recognizes the dual role of motion constraints. Indeed, it is recognition of this duality that has made it preferable to use the term constraint force rather than reaction. A primary benefit of the Lagrangian formulation is the ability to automatically account for constraint forces in the equations of motion. The formulation will allow us to treat connected bodies as a single system, rather than individual entities. The primary kinetic quantity for Lagrange's equations of motion is mechanical energy (kinetic and potential), whereas the Newtonian equations of motion are time derivatives of momentum principles.

The term analytical mechanics, which encompasses the developments of Lagrange, Hamilton, and many others who followed Euler, refers to the fact that the procedures that we shall develop are more mathematical than those of Newton and Euler. They also are more abstract. In fact, we often will find that features of the equations of motion, as well as of the physical responses predicted by those equations, are most readily explained in terms of Newton–Euler concepts.