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

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

d) ,,,,, ,,,,,,,, L 5 2

d) ,,,,, ,,,,,,,, L 5 2 248 3f \ 2 A 4 (graph of the time-function - section by ql =0) e) 3 5 3 H 4 (graph of the time-function - section by ql =0) Fig.6,7 Generating families of generic symplectic images for an obstacle surface in I~ 3. Remark 6.5 By choosing the special symplectic structure fibred over (Pl,P2)in the A3(H 3) case, we can investigate only a cuspidal edge of (I~. A) (L). In fact its generating family is F 1 (~,,~t,p) = F2(X,t,t ) - ~tlP 1 - I.t2P 2 , and after reduction of the parameters ~tl,ii 2 and X 2 we obtain the generating family for the H 2 singularity, 1 2 F12 (~.,p) = 9 ~5 _ p2k 3 +2" p27~" Its level-sets (wavefronts) have been written down in Table 2 of Scherbak [1988]. This observation is connected with a much more general feature of singular wavefront evolutions at an obstacle. Namely all singularities in obstacle geometry indicated in Table 2 of Scherbak [19881 are generated by generalized open swallowtails (in (t3,~) space) with the following generating families (see Janeczko [1986] p. 106): ~" k+l A2(k+l): J ( xk+l ~ ,2 . + i~a Si_lXk_i+l) dx ~'b- (~ > 1) and A b- (~ -> 2) singular wavefront evolutions are reconstructed from

249 A2(k+l) singularities by specifying appropriate common genetic positions of A c_ H and /(2(k+1) c_ (lVi,aS). This fact was obtained in another way in Givental [1988] by classifying the stable projections of the open swallowtails. Appendix 1. Classical Mechanics in Universal Phase Space Let Q be a manifold of dimension n with coordinates (qi), representing the configuration space of a mechanical system. The cotangent bundle T*Q of covariant vectors with coordinates (qi,Pj) represents the usual phase space of the system, Abraham and Marsden [1978]. Trajectories of the system in phase space are described by functions qi(0, pj(t), and the time-evolution of the system over an interval It, t'] is given by equations qi(t') = f~t'O(q(0, p(t)) (A. 1) pj(t') = ~t'O(q(t), P(O). Equations (A.1) represent a canonical transformation of T'Q, i.e. i~d (t,t') d f!t,t3, gi "(q'P) ^ j tq,P) = XdPi, ^ dq i. If the boundary values (q(t'), q(t') ; q(t), p(t)) are interpreted as coordinates in the space P = T*Q x T'Q, then equations (A.1) represent a submanifold R(t,t') c_ p of dimension 2n, which obviously is Lagrangian, endowed with the form to = XdPi(t') ^ dqi(t' ) - XdPi(t) ^ dqi(t ) . I 1 (P, co) is the universal phase space of classical mechanics. Take the special fibration represented by the l-form 0 = X(Pi(t')dqi(t') - Pi(t)dqi(t)), 1 such that c0 = dO. We can represent R (t,t') by a generating function. If R (t,t') is parametrized by (q(t'), q(t)) then there exists a function G(t,t')(q(t'), q(t)) such that R(t,t') is described by the equations equivalent to the system X(Pi(t')dqi(t') - Pi(t)dqi(t)) -- dG(t,t')(q(t'), q(t)), 1 OG(t.t3 0G(u') Pi(t')=Oq~'PJ (t)= 0qj(0 " This function is the Hamilton principal function, Tulczyjew [1974].

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