In this chapter we will explore three more stages in the nineteenth-century development of dynamics. One program was not itself directed at finding new foundational posits for the theory. Its initial purpose, rather, was to supply a method to facilitate the solving of dynamical problems, especially when they were framed in the mode of the Hamiltonian dynamical equations. We need to pursue it a bit, however, since it provides some of the resources needed to understand the second program treated in this chapter.

This second program is Hamilton–Jacobi theory. Here, starting from Hamilton's work in optics, rather than in dynamics, the result was the development of new possible foundational equations for dynamics to supplement those already known. Just as in the case of the development of the Hamilton dynamical equations, there was no claim here that the results went beyond the existing foundational posits in any fundamental way. It was universally accepted that the existing foundational methods were correct and complete as they stood. Rather, a new “reformulation” of the existing foundations was what was on offer.