HAMILTON, William Rowan. On a General Method in Dynamics; by which the Study of the Motions of all free Systems of attracting or repelling Points is reduced to the Search and Differentiation of one central Relation, or characteristic Function. [with:] Second Essay on a General Method in Dynamics.
London: Richard Taylor, 1834-35. First edition.

“Hamilton’s first general statement of the characteristic function applied to dynamics was his famous paper ‘On a General Method in Dynamics’ (1834)” (D.S.B. VI, p.88). The analogy between geometrical optics and mechanics, which Hamilton established in this paper, plays a fundamental role in all of modern physics, and was the basis of Erwin Schrödinger’s formulation of wave mechanics. Hamilton’s characteristic (or principal) function [“the first of his two great discoveries, the second was the quaternions” (DSB)] originated from his earlier work on geometrical optics. He introduced this function in an earlier paper (Theory of Systems of Rays, 1827) as a way of characterizing systems of rays of light being reflected by mirrors. “Hamilton's initial motivation was to cast optics into a scheme having ‘the power and dignity ... of the general method of Lagrange’ in mechanics, ... The main tool of his approach in optics had been what he called the characteristic function V, which he had connected with the principle of least action. This tool of the characteristic function could also be applied, ..., to reformulate the fundamental laws of dynamics; thus the actual motion of mass point in a field of forces, e.g., is found to be governed by equations that are the analogues of those determining the propagation of the rays of light. The particular extension of the optical methods led the author to a new ‘General Method in Dynamics,’ the Hamiltonian scheme, which is based on Hamilton’s variational principle of least action, the characteristic or principal function, and Hamilton’s (canonical) equations of motion. Hamilton provided his new general method of dynamics in two memoires, which were published in the Philosophical Transactions” (Mehra & Rechenberg: The Historical Development of Quantum Theory, vol. 5, p.511). Hamilton’s optical-mechanical analogy, not only provided a new and more powerful formulation of classical mechanics but also, came to form the foundation of the Schrödinger scheme of quantum mechanics, e.g., wave mechanics. “Hamilton introduced the methods of geometrical optics into mechanics and obtained an equation similar to the iconal equation and now known as the Hamilton-Jacobi differential equation. In it the index of refraction is replaced, essentially, by the potential energy and mass of the mechanical particle. In Hamilton’s work Schrödinger thus found an analogy between mechanics and geometrical optics. And, since geometrical optics ‘is only a gross approximation for light,’ he conjectured that the same cause was responsible for the failure of classical mechanics ‘in the case of very small orbital dimensions and very strong orbital curvature.’ Both would be only approximations for small wavelengths. Therefore, he said: ‘Perhaps this failure is a complete analogy to the failure of geometrical optics, that is, the optics with infinitely small wavelengths; (a failure) that occurs, as is known, as soon as the obstacles or openings are no longer large relative to the real, finite wavelength. Perhaps our classical mechanics is the complete analogue of geometrical optics and. as such, false... . Therefore, we have to seek an undulatory mechanics—and the way to it that lies closest at hand is the wave-theoretical elaboration of Hamilton’s model.’ Consequently, Schrödiger introduced into his development of wave mechanics conceptions that differed completely from those underlying the quantum mechanics formulated by the Göttingen school.” (D.S.B. article for Schrödinger).

Extracted from the Philosophical Transactions of the Royal Society, Part II, 1834, pp. 247-308; Part I, 1835, pp. 95-144. Each part stapled into early 20th century stiff wrappers, bookseller’s ticket of Henry Sotheran on upper cover. Fine and clean. 4to: 299 x 238 mm, uncut.