Friday, August 16, 2013
Hamilton Papers on Optics
Hamilton's paper on the Paths of Light and the Planets was more in the nature of a review of what he had already done and may have been intended for a broader audience than those interested in mathematical optics. The presentation appears to be overly simplistic and he may have caved in a little to his publishers. The first part of his paper was read in 1824 and it was not until 1828 that it was published in the Transactions of the Royal Irish Academy. The original plan was for three parts but time delays led him to modify this by publishing three additional supplements.
Hamilton's Theory of Systems of Rays
1st part, 1st supplement, 2nd supplement, 3rd supplement
Some of the topics mentioned were surfaces of constant action, surfaces and perpendicular rays and their relation to the characteristic function, the use of the elliptic integral of the 1st kind as a small angle approximation for the intensity of light near the focus when there is aberration present. He also gave the partial differential equation for the characteristic function with its dependence on the index of refraction. Hamilton also computed the 24 coefficients needed for extraordinary refraction.
Some loose ends:
1.) There are other uses for the elliptic integral of the 1st kind. It can be used for example to compute the nonlinear motion of a pendulum.
2.) The problem of determining the distance of a point from a plane using the length of the edges of rectangular triangular pyramid is an example of a set of covariant coordinates. One can think of them as the normal distances from the planes parallel to the coordinate planes through the point. The terms covariant and contravariant were introduced by Sylvester in a paper on Algebraic Forms publish in 1851. Covariant and contravariant coordinates are used in General Relativity.
3.) Extraordinary refraction involves crystals whose properties depend on direction and therefore involve tensors.
Hamilton clearly was ahead of his time and help point people in the direction that led to our current understanding of nature.