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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

254 dy for both cases.

254 dy for both cases. We find that ~- and d2y --~ are equal, and d3y 1- 0 24 d3y [0 1 ~3 v ~ = dx 3 - 0 -- - 32(t_1)3 ' (t_l)3 (t~ 1), so that the wavefronts are tangent to second order but not to third along the line t in Fig. A.4. References Abraham, R. and Marsden, J.E.[1978]. Foundations of Mechanics, (2nd ed.), Benjamin, Reading. Alexander, S.B., Berg, I.D. and Bishop, R.L. [1987]. The Riemannian obstacle problem, Illinois J. Math. 31 167-184. Arnold, V.I. [1981]. Lagrangian manifolds with singularities, asymptotic rays, and the open swallowtail, Funct. Anal. Appl. 15 235-246. Arnold, V.I. [1983]. Singularities in the variational calculus, Itogi Nauki, Contemporary Problems in Math. 22 3-55. Arnold, V.I. and Givental, A.B., Symplectic geometry [1985]. Itogi Nauki, Contemporary Problems in Math., Fundamental directions 4 5-139. Arnold, V.I., Gusein-Zade, S.M, and Varchenko, A.N. [1985]. Singularities of Differentiable Maps vol I, Birkhiluser, Boston. Berry, M.V.[1981]. Regularity and chaos in classical mechanics, illustrated by three deformations of a circular "oilliard', Eur. J. Phys. 2 91-102. Chernoff, P.R. and Marsden, J.E. [1974]. Properties of infinite dimensional Hamiltonian systems, Lecture Notes in Math. 425, Springer, Berlin. Dangelmayr, G. and Gtittinger, W. [1982]. Topological approach to remote sensing, Geophys. J.R. Astr. Soc. 71 79-126. Dirac, P.A.M. [1950]. Generalized Hamiltonian Dynamics, Canad. J. Math. 2 129-148. Guillemin, V. and Sternberg, S. [1984]. Symplectic Techniques in Physics, Cambridge Univ. Press, Cambridge. Givental, A.B. [1988]. Singular Lagrangian manifolds and their Lagrangian mappings, Itogi Nauki, Contemporary Problems in Mathematics 33 55-112. Golubitsky, M., Schaeffer, D.G. [1979]. A theory for imperfect bifurcation via singularity theory, Comm. Pure Appl. Math. 32 21-98. Janeczko, S. [1986]. Generating families for images of Lagrangian submanifolds and open swallowtails, Math. Proc. Camb. Phil. Soc. 100 91-107. Janeczko, S. [1987]. Singularities in the geometry of an obstacle, Suppl. ai Rend. del Circolo Matematico di Palermo (2nd ser.) 16 71-84. Keller, J.B. [1978]. Rays, waves and asymptotics, Bull. Amer. Math. Soc. 84 727-749. Luneburg, R.K. [1964]. Mathematical Theory of Optics, Univ. of California Press, Berkeley.

255 Poston, T. and Stewart, I. [1978]. Catastrophe Theory and its Applications, Pitman, London. Rychlik, M.R. [1989]. Periodic points of the billiard ball map in a convex domain, J. Diff. Geom. 30 191-205. Scherbak, O.P. [1988]. Wave fronts and reflection groups, Uspekhi Mat. Nauk 43 125- 160. Sniatycki, J., and Tulczyjew, W.M. [1972]. Generating forms of Lagrangian submanifolds, Indiana Math. J. 22 267-275. Sommerfeld, A. [1964]. Thermodynamics and Statistical Mechanics, Academic Press, New York. Tulczyjew, W.M. [1974]. Hamiltonian systems, Lagrangian systems and the Legendre transformation, Symposia Mathematica, 14 247-258. Weinstein, A. [1978]. Lectures on symplectic manifolds, C.B.M.S. Conf. Series 29, Amer. Math. Soc., Providence. Weinstein, A. [1981]. Symplectic geometry, Bull. Amer. Math. Soc. 5 1-13. Zakalyukin, V.M. [1976]. On Lagrangian and Legendrian singularities, Funct. Anal. Appl. 10 23-31.

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