Book contents
- Frontmatter
- Contents
- Chapter 1 Introduction
- Chapter 2 The pre-history of classical dynamics
- Chapter 3 The astronomical revolution
- Chapter 4 Precursors to Newtonian dynamics
- Chapter 5 The Newtonian synthesis
- Chapter 6 Philosophical aspects of the Newtonian synthesis
- Chapter 7 The history of statics
- Chapter 8 The development of dynamics after Newton
- Chapter 9 The “Newtonian” approach after Newton
- Chapter 10 From virtual work to Lagrange's equation
- Chapter 11 Extremal principles
- Chapter 12 Some philosophical reflections on explanation and theory
- Chapter 13 Conservation principles
- Chapter 14 Hamilton's equations
- Chapter 15 Canonical transformations, optical analogies and algebraic structures
- Chapter 16 The search for new foundations
- Chapter 17 New directions in the applications of dynamics
- Chapter 18 Spacetime formulations of Newtonian dynamics
- Chapter 19 Formalization: mass and force
- Chapter 20 Relationist dynamics
- Chapter 21 Modes of explanation
- Chapter 22 Retrospective and conclusions
- References
- Index
Chapter 16 - The search for new foundations
Published online by Cambridge University Press: 05 December 2012
- Frontmatter
- Contents
- Chapter 1 Introduction
- Chapter 2 The pre-history of classical dynamics
- Chapter 3 The astronomical revolution
- Chapter 4 Precursors to Newtonian dynamics
- Chapter 5 The Newtonian synthesis
- Chapter 6 Philosophical aspects of the Newtonian synthesis
- Chapter 7 The history of statics
- Chapter 8 The development of dynamics after Newton
- Chapter 9 The “Newtonian” approach after Newton
- Chapter 10 From virtual work to Lagrange's equation
- Chapter 11 Extremal principles
- Chapter 12 Some philosophical reflections on explanation and theory
- Chapter 13 Conservation principles
- Chapter 14 Hamilton's equations
- Chapter 15 Canonical transformations, optical analogies and algebraic structures
- Chapter 16 The search for new foundations
- Chapter 17 New directions in the applications of dynamics
- Chapter 18 Spacetime formulations of Newtonian dynamics
- Chapter 19 Formalization: mass and force
- Chapter 20 Relationist dynamics
- Chapter 21 Modes of explanation
- Chapter 22 Retrospective and conclusions
- References
- Index
Summary
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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- Philosophy and the Foundations of Dynamics , pp. 144 - 169Publisher: Cambridge University PressPrint publication year: 2012