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Energy, the Subtle ConceptThe discovery of Feynman's blocks from Leibniz to Einstein$
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Jennifer Coopersmith

Print publication date: 2015

Print ISBN-13: 9780198716747

Published to Oxford Scholarship Online: August 2015

DOI: 10.1093/acprof:oso/9780198716747.001.0001

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Hamilton and Green

Hamilton and Green

Chapter:
(p.226) 13 Hamilton and Green
Source:
Energy, the Subtle Concept
Author(s):

Jennifer Coopersmith

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198716747.003.0013

The unknown George Green, from Nottingham, self-published An Essay on the Mathematical Analysis of Electricity and Magnetism, in which the potential function, V, was defined for the first time. The mathematical prodigy William Rowan Hamilton, inspired by Descartes, described the geometry of light rays algebraically. He extended this method to the motion of particles—his optico-mechanical theory—showing that particles exhibited wave features, but now ‘action’ was minimized instead of time. When a dual system of generalized coordinates (q and p) was used, and when T in the Lagrangian was normalized to a special form, a new energy function emerged—the Hamiltonian, H. The form of the Hamiltonian determined the evolution of the given mechanical system. The ‘canonical’ coordinates p and q occurred in an imaginary (virtual) phase space, and mapped out the range of mechanical possibilities. The links to Poisson brackets and Schrodinger’s wave mechanics are mentioned.

Keywords:   George Green, William Rowan Hamilton, potential function, the Hamiltonian, phase space, canonical coordinates, Poisson brackets, Schrodinger’s wave equation, Carl Jacobi, q and p

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