The Legendre transformations in Hamiltonian optics

Andrey V Gitin

doi:10.2971/jeos.2010.10022

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.

Highlights

The best way to solve a problem is to look at it from the right point of view
It is very important for everybody to have a possibility to see the problem from various points of view. In physics this possibility is given by one-to-one correspondences, for example the Fourier transformations in Fourieroptics and the Legendre transformations in Hamiltonian optics
Applying Euler’s equations, Eqs. (8), we find that light rays are the solutions of Hamilton’s equations [4]

Summary

INTRODUCTION

The best way to solve a problem is to look at it from the right point of view. It is very important for everybody to have a possibility to see the problem from various points of view. In physics this possibility is given by one-to-one correspondences, for example the Fourier transformations in Fourieroptics and the Legendre transformations in Hamiltonian optics. Note that in Hamiltonian optics the Legendre transformations appear twice: as a connection between the Lagrangian and the Hamiltonian and as relations among eikonals. These types of the Legendre transformations are used independently.

FERMAT’S PRINCIPLE AS THE BASIS OF HAMILTONIAN OPTICS

THE LEGENDRE TRANSFORMATIONS

THE CONNECTION BETWEEN TWO TYPES OF LEGENDRE TRANSFORMATIONS

ANALOG OF EULER’S EQUATIONS IN MOMENTUM REPRESENTATION

CONCLUSIONS