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

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

194 is equivalent to its

194 is equivalent to its versality as an unfolding. Hence associated to each organising centre (Remark 1.3) there is an essentially unique stable Morse family. This is no longer true for equivariant Morse families - thus the unfoldings themselves must be classified, not just the organising centres. In § 1 we give a brief description of the theory of generating functions and Morse families for G-invariant Lagrange submanifolds in the neighbourhood of a fixed point, x 0, of the G action on X. We also show that generic invariant caustics do not pass through isolated fixed points (Proposition 1.2). Equivariant symplectic equivalence is defined in §2, and then translated into an equivalence relation between invariant Morse families. We describe the tangent space for this equivalence relation and discuss infinitesimal stability. When G is finite, generic Morse families can be pulled back from G-versal families, in the sense of Slodowy IS]. In this case stability is equivalent to the stability of the pull back mapping with respect to equivariant right equivalence (Proposition 2.7). In §3 we adapt the finite determinacy estimates of Bruce, du Plessis and Wall [BPW] to the present context. These use special properties of unipotent algebraic groups to obtain the best possible estimates. Finally in §4 we apply the ideas of the previous sections to the classification of stable, corank 1, E2-invariant Lagrange projections. Theorem 4.5 gives complete classifications for dim X < 7, for the case when dim Fix(Z2,X) < ~dim X, and for the case dim Fix(Z2,X) = dim X - 1. We also show that if dim Fix(l 2,X) > ½ dim X then there exist generic 12-equivariant Lagrange projections which are not stable. Symmetric caustics arise naturally in a number of contexts - the principle motivation for our work is provided by the following examples. Optical systems. In IN], Nye gives a detailed experimental and theoretical investigation of the caustics that are obtained by refraction of light through a drop of water with the symmetry of a square. Nye's analysis of the possibiities is based on a detailed study of a particular unfolding of the singularity X 9. However it stops short of a rigorous classification. Phonon focusing in crystals. At very low temperatures crystals conduct heat "ballistically" - ie without diffusion or scattering [W]. A heat pulse applied at one face of a crystal propagates anisotropically and is channeled along certain crystal directions. By modelling this propagation as a wave moving through an elastic medium, the locus of points of maximum heat intensity can be interpreted as a caustic lAD]. Phasc transitions in crystals. The magnetic phases of a ferromagnetic crystal can be modelled as critical points of a "free energy" function, F(p,q), which depends on both

195 internal variables p and external variables q such as temperature and applied magnetic fields IE]. The critical points are taken with respect to p and the variables q are regarded as parameters. The point symmetry group of the crystal acts on both the internal and external variables and F must be invariant under these actions. The locus of phase transitions can be identified with the local bifurcation set of F, and is therefore a caustic. Other types of phase transitions can be treated in the same way, as indeed can general symmetric gradient bifurcation problems in which the symmetry group acts on both the internal state variables and the external parameters. Acknowledgements The work of Staszek Janeczko was partially supported by SERC and by the Max- Planck-Institut fur Mathematik. The work of Mark Roberts was supported by a SERC Advanced Research Fellowship.

Reading grade 6 2.A.5.b - mdk12
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