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

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

230 torsion spring of

230 torsion spring of unit strength. The angle tp and the force pq are coordinates of a manifold X. Together with the torque pq) and position q they form a coordinate system (9,Pq,Ptp,-q) on T*X. The potential energy of this system, that is the generating function for the Lagrangian submanifold L c T'X, has the form F(9,pq) = ½9 2 - 2pqCOS cp. In the reduced phase space T*Y with local coordinates (pq,-q) and the reduced mapping P(9,Pq) = Pq we obtain for the image of L: T*p(L) = {(pq,-q) • T*Y : 3cpe[0,2n), cp+2pqsin q) = 0, q = 2 cos tp}. It has a singularity at the point (pq,q) = (-~,2). Generation of New Symplectic Manifolds Let (P,to) be a symplectic manifold. Let H be a hypersurface in P (say the zero- level set of the Hamiltonian function) or more generally a submanifold of P with the coisotropyproperty, i.e. at each point p • H Vp = {v • TpP : to(v,u) = 0 V u ~ TpH} ~_ TpH. Characteristics (see Abraham and Marsden [1978] p. 82) of the distribution Vp form a foliation of H, the foliation by integral curves of the Hamiltonian system. Let M denote the set of these characteristics and let m be the canonical projection along characteristics, rc :H~M. Then R c_ p × M, R = graph(n) is a symplectic relation from P to M with uniquely defined symplectic structure 13 on M, :x*[$ = col H . Examples 3.3 1 The space of binary forms 2k+3 2k+2 k+2 k+l r x x y x y P = ~-q0~ + q l ~ + "'" + qk+l (k+2)! k+l k+2 1 x y (_ 1)k+2p0y2k+3 } +(-1) Pk+f (k+l)! ~"'+ is endowed with the unique $12(~)-invariant symplectic structure n+I to -- i=~dPi^dqi. Let the coisotropic submanifold C in P be defined by G(p,q) = q0-1 and the Hamiltonian of translations of ×, +~ 2 H(p,q) = p l+qlP2 + ... + qkPk+! gqk+l' {G,H} --- 0, C = {(p,q) • P: G(p,q) = O, H(p,q) = 0}.

231 Then the reduced symplectic space is the space of polynomials 2k+l 2k- l k k- l x x x x (_l)kpl} C 7t ) ~ = { ~ + q l ~ + ... + qk~.l - Pk(k_~. + "'" + with the reduced symplectic structure k = ~ dPi^dqi" This symplectic structure appeared in Arnold [1981, 1983] as a model for the investigation of singularities of systems of rays in various variational problems. See also Jancczko [1986] p.103. 2 Reduction in systems with symmetry. Let H be a Hamiltonian invariant under a group action. The process of Marsden-Weinstein reduction defines a symplectic structure on orbit-spaces of momentum level-sets of H. At a singular point of the momentum mapping the orbit-space may have singularities. See Abraham and Marsden [1978] p.298. Stable Lagrangian Projections To describe the mutual positions of the Lagrangian submanifold L ~ T*X and the nontransverse fibre of T*X we introduce the notion of fibre equivalence of germs of Lagrangian submanifolds or their Lagrange projections ~XOiL: L ~ X where iL:L ~ T*X is an immersion of L, see Arnold and Givental [1985]. Let (L 1,Pl) and (L2,P2) be two germs of Lagrangian submanifolds of T*X. They are fibre equivalent if there exists a germ of a symplectomorphism O:(T*X,p 1) -~ (T*X,P2) preserving the fibre structure nx:T*X -~ X, and such that O(L 1) = L2, O(Pl) = P2" Introduce the Whitney C oo topology in the space of Lagrange immersions iL: L ~ T*X. Now we say that a Lagrangian submanifold L 1 c T*X is close to L if the corresponding immersion iLl :LI~ T*X is close to i L, that is, iLl belongs to some open neighbourhood of i L in the Whitney topology. The Lagrangian germ (L,p) c T*X is Lagrange stable if for any sufficiently close Lagrangian submanifold L 1 c T*X there exists a point p 1 ~ T*X close to p, such that (Ll,Pl) and (L2,P2) are equivalent. Arnold and Givental [1985] show that Lagrange stability is a generic property for dim X < 5. All 16 stable models, including the trivial one, are classified in Arnold, Gusein-Zade and Varchenko [1985]. They appear in the respective dimensions as follows: dimX=l: A 1,A 2. dim X = 2: A 1, A 2, A3 +. dimX--- 3: A1, A2,A3+,A4, D4 ±.

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