The preceding developments suffice to treat systems that are described by a finite number of degrees of freedom. They are not directly applicable if a system is best modeled as a flexible continuum, in which bodies deform and also have mass. One cannot compartmentalize kinetic and potential energy in such systems, because a mass element also stores strain energy. Consequently, concepts like generalized coordinates become problematic. The derivation of principles that can be used to model continuous media is the first priority for this chapter.

Another focus here is exploration of alternative formulations for deriving equations of motion for discrete systems. Derivation of these formulations has received considerable attention for more than a century and a half. Those efforts were motivated by a desire to seek simpler equation forms, either from the perspective of ease of implementation or ease of solution. We consider a few formulations, but extensive discussions may be found in the works by Greenwood (2003) or Papastavridis (1998, 2002).

One of the outcomes of these alternative formulations are conservation principles that sometimes can be used when the standard momentum and energy principles cannot be implemented. Such principles enable us to determine features of a system's response without solving equations of motion and also provide checks for computation solutions. Overall, the developments that follow are intended to enhance understanding of the basic concepts of analytical mechanics and to provide increased versatility to carry an analysis to completion.