W. R. Hamilton, Trans. R. Irish Acad. 15, 69 (1828); 16, 1(1830); 16, 93 (1831); 17, 1 (1837). Reprinted in The Mathematical Papers of Sir W. R. Hamilton, Vol. I, Geometrical Optics, A. W. Conway and J. L. Synge, eds. (Cambridge U. Press, Cambridge, 1931).

See, for example, M. Born and E. Wolf, Principles of Optics, 2nd ed. (Macmillan, New York, 1964), pp. 134, 211, 221.

Matrix methods are described, for example, by K. Halbach, Am. J. Phys., 32, 90 (1964); M. Klein, Optics (Wiley, New York, 1970), p. 84; W. Brouwer, Matrix Methods in Optical Instrument Design (Benjamin, New York, 1964).

H. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968); An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, 1970), p. 36. See also the following papers of H. Buchdahl: J. Opt. Soc. Am. 62, 1314 (1972); Optik 37, 571 (1973); 40, 460 (1974); 46, 287 (1976); 46, 393 (1976); 48, 53 (1977).

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1953), Vol. 1, pp. 164–190.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 15.

There does not seem to be any particularly transparent derivation of this condition in the literature. However, it may be inferred, for example, from H. Goldstein, Classical Mechanics, 2nd ed. (Addison-Wesley, Reading, Mass., 1980), pp. 394–396; E. Salatan and A. Cromer, Theoretical Mechanics (Wiley, New York, 1971), p. 222.

H. Weyl, The Classical Groups (Princeton U. Press, Princeton, N.J., 1946), p. 165.

O. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972), p. 245. This book contains an extensive discussion of optics from a group theoretical perspective. The reader is also referred to the paper of M. Herzberger, Trans. Am. Math. Soc. 53, 218 (1943), for an early discussion of the consequences of what is essentially the symplectic condition.

A. Dragt and J. Finn, J. Math. Phys. 17, 2215 (1976).

It is not necessarily the case that every symplectic map corresponds to a physically realizable optical system.

A. Ghatak and K. Thyagarajan, Contemporary Optics (Plenum, New York, 1978), p. 52.

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