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

## TL;DR

This paper explores William Rowan Hamilton's contributions to geometrical optics, focusing on the Malus-Dupin theorem, its proofs, and its implications for optical focusing and symplectic geometry.

## Contribution

It presents Hamilton's original and symplectic proofs of the Malus-Dupin theorem, highlighting its significance in optics and Hamilton's development of the characteristic function.

## Key findings

- The Malus-Dupin theorem characterizes focusing conditions for light rays.
- Hamilton's proof connects optics with symplectic geometry.
- The theorem's invariance under reflections and refractions is established.

## Abstract

The works of William Rowan Hamilton in Geometrical Optics are presented, with emphasis on the Malus-Dupin theorem. According to that theorem, a family of light rays depending on two parameters can be focused to a single point by an optical 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 a rectangular system of rays remains rectangular after an arbitrary number of reflections through, or refractions across, smooth surfaces of arbitrary shape. The original proof of that theorem due to Hamilton is presented, along with another proof founded in symplectic geometry. It was the proof of that theorem which led Hamilton to introduce his \emph{characteristic function} in Optics, then in Dynamics under the name \emph{action integral}