London: Richard Taylor. 1st Edition. FIRST EDITIONS OF HAMILTON'S FAMOUS TWO 'ESSAYS' ON DYNAMICS, SEMINAL WORKS IN WHICH HE DEVELOPS THE WHOLE OF THEORETICAL DYNAMICS BY THE AID OF ONE CENTRAL FUNCTION, HIS 'CHARACTERISTIC' (or 'principal') FUNCTION.
"The analogies Hamilton establishes in these papers between geometrical optics and mechanics would go on to play a fundamental role in all of modern physics and provide the basis of Schrodinger's formulation of wave mechanics" (Sophia Rare Books). In the first essay, Hamilton "defines his characteristic function by analogy with his researches in optics, and develops its chief properties for a general system of points in any system of coordinates. The remainder of the paper is devoted to methods of approximation with a view to applying them to perturbations of astronomical bodies. At the end he introduces another function, the [characteristic or] principal function, which he develops in the second essay.
[In the second essay] Hamilton develops the properties of the principal function in much the same way as in the previous Essay, but here established for the first time his well-known equations of motion. He applies his method to a case of planetary motion, using a system of canonical elements" (Introduction, The Mathematical Paper of Sir William Rowan Hamilton, xiii). Hamilton then argues that the "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. 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'"(Sophia Rare Books). Item #909
CONDITION & DETAILS: London: Richard Taylor. 4to. Extracted from the Philosophical Transactions of the Royal Society, 1834 and 1835. 11 x 8.75 inches; 12 x 9.25 inches (the volumes these were extracted from were slightly differently sized). Part I: pp. 247-308; Part II: 25-144. Complete. Clean and bright. Near fine condition in every way.