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

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

## 226 2 Hamiltonian

226 2 Hamiltonian dynamics of a particle system may be considered as a Lagrangian submanifold L of (TP,to), defined in Example 2.2.2. If H:P --} ~ is the Hamiltonian then n tol L = d[ lbidqi -/lidPi) = -dH(q,p)) = 0, where H is a generating function for L. In other words the dynamics of a particle system in the potential U defines a Lagrangian submanifold • a L = {(q,P,/t,lb) ~ TP : Pi = mcli, Pi = - ~qi U(q)}. 3 Partial differential equations of field theory can be viewed as Lagrangian submanifolds. Let D be a domain in ~ 3 and let X = (p,tp), Y = (p,t~), where cp,t~ are functions on OD (Dirichlet boundary conditions) and p,~ are measures on 3D of the form 3 p = i_~tPidOi where d6 i is a standard measure on 0D. For X, Y in the space P of such Dirichlet/Neumann boundary conditions on 0D we may introduce, as in Chemoff and Marsden [1974], a canonical symplectic structure f~(X,Y) = JD (p.t~ - ~.{p). We assume tp, {~ ~ H½(OD), and p, ~ ~ H -½ (OD). By the limit procedure D --} x we obtain the new symplectic form to in the appropriate space of jets of mappings x D (9(x), Pi(X)) at x, defined by to = dR ^ dqo + i~l= dPi ^ dtPi R = 3 ~ 3Pi _ i = 1 dx i 3 q~i=Txi , i=1,2,3. Thus the Poisson equation A 9 = f in ~q3 determines a Lagrangian submanifold L = {(tp,pi, tpt ~, R) : Pi = tPi, R = f}. Lagrangian varieties In classical mechanics with constraints, or in thermodynamics, there appear Lagrangian subsets modelling the spaces of equilibrium states, see Janeczko [1986]. They are no longer smooth but still have Lagrangian properties. We define a Lagrangian variety in symplectic space (P,to) to be a stratifiable subset of P with all strata isotropic

227 according to co and all maximal strata Lagrangian. We see that in this sense all semialgebraic curves in 2-dimensional symplectic space are Lagrangian varieties. Examples 2.4 1 The set of rays (oriented lines in ~ 2) produced by the singular wavefront with a cusp singularity is the conormal bundle to the wavefront (Fig. 2.1). It is Lagrangian in (T*~ 2, o~t2) and its closure is singular with a so-called open Whitney umbrella singularity at 0, see Arnold [1981], Janeczko [1986]. Fig. 2.1 Ray system produced by a singular wavefront. n 2 Let (L,0) c (~12n, co = ~ dxi^dYi) be the set-germ defined as follows: i=l OF OF L= {x,y e ~2n: 3~.e~k, Yi = ~x~xi (x,~L), %(x,~L) - O, l

1O 9 I B I 7 I 6 5 4 I 3 2 1