The works of William Rowan Hamilton in geometrical optics and the Malus-Dupin theorem

摘要

The works of William Rowan Hamilton in Geometrical Optics are presented, withemphasis on the Malus-Dupin theorem. According to that theorem, a family oflight rays depending on two parameters can be focused to a single point by anoptical instrument made of reflecting or refracting surfaces if and only if,before entering the optical instrument, the family of rays is rectangular(\emph{i.e.}, admits orthogonal surfaces). Moreover, that theorem states that arectangular system of rays remains rectangular after an arbitrary number ofreflections through, or refractions across, smooth surfaces of arbitrary shape.The original proof of that theorem due to Hamilton is presented, along withanother proof founded in symplectic geometry. It was the proof of that theoremwhich led Hamilton to introduce his \emph{characteristic function} in Optics,then in Dynamics under the name \emph{action integral}