Journal of the European Optical Society - Rapid publications, Vol 5 (2010)

The Legendre transformations in Hamiltonian optics

A. V. Gitin


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.

© The Authors. All rights reserved. [DOI: 10.2971/jeos.2010.10022]

Full Text: PDF

Citation Details

Cite this article


R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley, 1964).

W. Brouwer, and A. Walther, "Geometrical optics" in Advanced Optical Techniques (North-Holland Publishing Company, Amsterdam, 1967).

M. Born, and E. Wolf, Principles of Optics (7th Edition, Cambridge University Press, Cambridge, 1999).

V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1989).

A. Walther, "Systematic approach to the teaching of lens theory" Am. J. Phys. 35, 808-816 (1967).

A. V. Gitin, "Using continuous Feynman integrals in the mathematical apparatus of geometrical and wave optics. A complex approach" J. Opt. Technol.+ 64, 729-735 (1997).

V. Guillemin, and S. Sternberg, Geometric Asymptotics (American Mathematical Society, Providence, 1977).

A. V. Gitin, "Legendre transformation: connection between transverse aberration of an optical system and its caustic" Opt. Comm. 281, 3062-3066 (2008).

A. V. Gitin, "The Legendre transformation as a way of constructing a phase portrait of beams of light rays" Opt. Comm. 282, 757-762 (2009).

A. Walther, "Radiometry and coherence" J. Opt. Soc. Am. 58, 1256- 1259 (1968).

K. H. Brenner, and J. Ojeda-Castañeda, "Ambiguity function and Wigner distribution function applied to partially coherent imagery" Opt. Acta 31, 213-223 (1984).

A. V. Gitin, "Optical systems for measuring the Wigner function of a laser beam by the method of phase-spatial tomography" Quantum Electron.+ 37, 85-91 (2007).

E. W. Marchand, Gradient Index Optics (Academic Press, New York, 1978).

L. D. Faddeev, and O. A. Yakubovskii, Lectures on Quantum Mechanics for Mathematics Students (American Mathematical Society, Providence, 2009).