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

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

228 Example 3.1 Let ¢p

228 Example 3.1 Let ¢p : P1 "} P2 be a symplectomorphism; then graph((p) c_ P1 x P2 is a symplectic relation; and R = {(p l,¢p(pl)) : Pl a P1 } n2*co 2 - ~,Xl*co 11R = q~*co2-col = 0. The notion of symplectic relation is a generalization of this example and it is quite useful in the symplectic category, see Sniatycki and Tulczyjew [1972]. Generating Functions In the above setting, let = ~2"co2 - ~l*C°l • Locally, we may write f~ in the form [2 = -dO; for example we could set O -- ~2"02- ~1"01 where-d0j = coj forj = 1,2. Let (P:P1 "} P2- Then i*~0dO = di*q~O = 0, that is, i*q~O is closed if and only if (p is symptectic. Locally there exists a function G:graph(cp) --} St such that i*q~O --- -dG. We call G a generating function for ¢p. It depends on the choice of O and is locally defined. See Abraham and Mardsden [1978] p.379 for more details. Symplectic Images Let S c_ (PI,COl) and let R be a symplectic relation. The image or pushout of S under R is the following subset of (P2,co2): R(S) = {P2 ~ P2 : 3Pl ~ S such that (Pl,P2) ~ R} If S' c (P2,CO2) then the preimage or pullback under R of S' is the image of S' under the transposed symplectic relation Rtc_ (P2 x PI ; tel*co 1 - n2*co2). Lagrangian Submanifolds of Cotangent Bundles as Symplectic Images Let p : X -, Y be a submersion between two manifolds. The symplectic lift T*p is the symplectic relation from (T'X, coX) to (T'Y, COy) given by T*p -- {(x,~ ; y,TI) ~ T*X x T*Y : y = p(x), ~i = ~-~PJ TIj}. J i In fact it is a symplectic reduction relation, see Janeczko [1986]. The nicest class of Lagrangian submanifolds in T*y is formed by those that are transverse to the fibres, i.e. given as the sections y P, dF(y) for some smooth function F:Y --> ~. It is known, see Weinstein [1978], that all Lagrangian submanifolds of (T'Y, COy), at least locally, are images of Lagrangian submanifolds in (T'X, COX) under symplectic reduction relations. More precisely, if L c_ ~*y, coy) is a Lagrangian submanifold then there exist:

a) a submersion p : X -~ Y, 229 b) a Lagrangian submanifold L c (T'X, co x) transverse to the fibres of T'X, i.e. L = graph(dF) for some function F : X ~ R, such that locally L = T*p(L). F is called a Morse family for L. In local coordinates p : R k x Y 4 Y is projection onto the second factor and (X 1 ..... X k) = X ~ Rk are the so-called Morse parameters, with ~2 F k < - dim Y, and rank( 3ZF, ~_._ .(k,y): k. dA,idx ~ Examples 3.2 1 Suppose that I: c T*IR is defined by f_, = [(p,q) : p2_q = O} in the usual coordinates (p,q). Then and SO 1 F : IR × IR -, N, F(k,q) -- ~- - -~,3-qZ, L : = {(~,p ; )k,q) ~ T2R 2 : la = K2-q, p = -;k) p(k,q) = q, T*p(L) = 1".. j//. I w° -- Fig.3.1 Geometric picture for Example 3.2. 2 Symplectic reduction of static mechanical systems, Abraham and Marsden [1978], usually leads to singular Lagrangian varieties. Consider for example the finite element analogue of the Euler beam, Golubitsky and Schaeffer [1979]. This system consists of two rigid rods of unit length subjected to a compressive force pq, which is resisted by a

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