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

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

272 Proposition 6

272 Proposition 6 Suppose (c, 4-) ¢ {(1, +), (-1, +)}, then f~ is 3.T~z2-determined and has eodimension ~. Moreover, is an TO+ 2-versal deformation of f~. PROOF: A simple computation shows that which, by intersecting with E3, implies Now apply Proposition 3. "d C mLJ(/), n EF c m+3.J-(f). Thus we have proved that every odd coraak 3 critical point has 7~z2-codimension at least 4. It follows that there does not exist a Z2 stable Lagrange map germ (R3,0) (R3,0). In fact any Z2 invariant Lagrangian germ (R3,0) --* (R3,0) has infinite codimen- sion, and so is not finitely T~+2-determined. The infinite codimension comes from the modulus c that occurs in the organizing centre: one can show that in general, if f is a non-simple germ and F any non-versal deformation of f, then the associated Lagrange map ~rF has infinite codimension. We therefore cannot hope to classify, or give a normal form for, generic Z2-invariant Lagrangian maps, at least not under smooth Lagrangian equivalence. There are two possible approaches to circumventing this problem. One is to use topo- logical Lagrangian equivalence, and the other is to use a weaker version of Lagrangian equivalence which S. Janeczko and M. Roberts call caustic equivalence in [8, 9]. We take the second approach. Caustic equivalence is designed to ensure that the caustics of caustic equivalent La- grangian maps are diffeomorphic. The definition is in terms of generating families using J. Damon's notion of Ev-equivalence which we define first; for more details, see [6]. Definition: Let gl,g2 : (Ra,0) -* (Rb,0) be two map germs, and V C (Rb,0) a subvariety germ. We say gl and g2 are/(:v-equivalent if there are diffcomorphisms H of (R" xR b, (0, 0)) and h of (R a, 0) such that . H(u, v) = (h(u), O(u, v)), for some map #, * H(R axV)=R'xV, e H(u,gl(u)) = (h(u),g2oh(u)). Remark In our application, V is not analytically trivial at the origin, i.e. every analytic flow preserving V fixes 0, which implies that #(u, 0) = 0 and K:v is a geometric subgroup of [61. El

273 Now we return to generating families. Let F1,F2 : (R" x R",0) --* R be deformations of the function germ f. Let ~ : (R" x R b, 0) ~ R be a'versal deformation of f, and let V C (R b, 0) be the discriminant of this deformation. Each Fi is induced from Y by a map gl : R a --* R b. Note that the caustic of 7tEl is the set g~'l(v). We say FI and F2 are caustic equivalent if the map germs gl and g2 are/Cv-equivalent. (We are being sloppy: F/is not necessarily induced from .T, but it is equivalent to a generating family which is induced from ~'. Since equivalent generating families define equivalent Lagrange maps, this sloppiness is unimportant.) Theorem 7 Let f~ be given by (1), and ~ its versal deformation given in (2). Suppose F : (R a × R a, O) ~ (R, O) is a deformation of f~, for some (c, +) satisfying, (e,-b) ¢ {(1,+),(-1,+),(,/5/2,+),(-,/5/2,+)}, such that the map g : (R 3, 0) --* (R 4, 0) inducing F from ~r is tran~erse to the t.axis, then F is caustic equivalent to the generating family, The proof of this result is deferred to the final section. - - q y - (3) We now proceed by describing the caustics of the generic invariant Lagrange projections which are physically allowable in classical Hamiltonian systems, see Remark 5. Let f~ be given by (1), with generic 3-parameter deformation, givcn by (3). The associated Lagrangian map germ is given by, r~c (x,y,z) = (3x 2 + 4cxz -b z2,2yz,2cx 2 -b 2xz). Recall from Theorem 7 that the exceptional values of (c,-b) are given by (c2,-b) = (1, +), (314 , +). Lemma 8 If [el > 1 or '+ = - ', then the image of r~ cannot be contained on one side of a smooth surface. PROOf: This is a straightforward calculation. First note that the restriction of Ir~ to the plane y = 0 maps to the plane q = 0. This restriction map is surjective in both the cases in the hypothesis. Moreover, the lines (0,-by, O) map to two line segments, one on each side of the p-r plane. The lemma follows. [] Thus we are left with ~r¢ = 7r + for [c[ < 1. The origin is a E3-polnt of 1re. There are E 2 points of 7r c near 0 if and only if c 2 = 3[4, which is excluded by hypothesis. (There are 3 real branches of ~2-points if e -- -v/3/2 and one if c = x/~/2, in the first case these are all hyperbolic umbilics, while in the second they are elliptic umbilics.) In this range of values of [e[, there are 3 branches of ~1,1 points which give rise to 3 cuspidal edges on the caustic. Further calculations show that the caustic is as drawn in Figure 2. The complement of the caustic has three components. Each point of the inner component has 8 preimage points; each point of the middle component has 4 points in its preimage, while the outer

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