Regular papers

The Legendre transformations in Hamiltonian optics

Max-Born-Institut f ür Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Str. 2a, D-12489 Berlin, Germany

^* agitin@mbi-berlin.de

Received: 25 November 2009

Abstract

The Legendre transformations are an important tool in theoretical physics. They play a critical role in mechanics, optics, and thermodynamics. In Hamiltonian optics the Legendre transformations appear twice: as the connection between the Lagrangian and the Hamiltonian and as relations among eikonals. In this article interconnections between these two types of Legendre transformations have been investigated. Using the method of “transition to the centre and difference coordinates” it is shown that four Legendre transformations which connect point, point-angle, angle-point, and angle eikonals of an optical system correspond to four Legendre transformations which connect four systems of equations: Euler’s equations, Hamilton’s equations, and two unknown before pairs of equations.