The nineteenth-century reconstructions of dynamics we have just surveyed were certainly not motivated by philosophical reflections on the traditional foundations of the theory. Nor were they formulated in response to any felt disquietude about the traditional theory. Hamilton's equations arise out of an understanding, developing at the time, that a single second-degree differential equation could be replaced by a pair of coupled first-degree equations. Hamilton–Jacobi theory has a less purely mathematical motivation, in that it followed from a deep understanding of the degree to which the formalisms of geometrical optics and of dynamics bore interesting parallels to one another. Here the primary inspiration comes from the understanding that the Principle of Least Time in the former theory and the Principle of Least Action in the latter were sufficiently similar that other formal similarities, such as advancing wave fronts and trajectories as rays orthogonal to these, might be found as well. The bracket formulation of dynamics is, once again, a purely formal manipulation of the theory.
Each of these reconstructions, as we have seen, has manifold consequences. They do play some role in extending the ability of the theory to be applied to difficult special cases. But they also provide deeper understandings of the hidden internal structures of the theory, as in, for example, the realization of the fascinating algebraic structure among the generalized variables revealed by the bracket notation or the “wave-front” structure in configuration space revealed by Hamilton–Jacobi theory. And they also provide just the resources that will later be needed in going beyond classical dynamics to newer theories, as in the application of Hamilton–Jacobi theory to the foundations of the wave-theory version of quantum mechanics and the application of the bracket notations in the formalizing of the matrix version of quantum theory.
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