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

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

224 defined at each

224 defined at each point p ~ P by a nonsingular 2n x 2n antisymmetric matrix Coij that depends smoothly on p. Examples 2.1 1 ~2n _~ ~n x ~n becomes a symplectic manifold with the symplectic form Co = ~dYiAdx i ! in coordinates (x,y) = (x 1,.-.xn,yl ..... Yn). 2 ~n is a symplectic manifold with the 2-form Im (k~__l dz k ® d~k). 3 The direct sum V* ~ V of a vector space V and its dual V*, endowed with the canonical form co : (~ 1 ~ Vl, ~2 ~ v2) ~ ~l(V2) - ~2(Vl) is a symplectic manifold. Let (PI,COl) and (P 2,co2) be symplectic manifolds of equal dimension. A symplectic map is a smooth map V:P 1 ~ P2 such that V'co2 = C°l. Here V* is the pullback V*CO(Pl,P2) -- CO(~(Pl),V(P2)). It follows (Abraham and Marsden [1978] p.177) that V is a local diffeomorphism, and we therefore refer to it as a local symplectomorphism. A global symplectomorphism v:P1 ~ P2 is a diffeomorphism that is symplectic, and if such a map exists we say that P1 and P2 are symplectomorphic. If Pl ~ P1 and P2 ~ P2 we say that P1 and P2 are locally symplectomorphic near p 1, P2 if there exists a symplectic map v:U 1 ~ U 2 where U 1 and U 2 are open, p 1 ~ U 1, P2 e U2, and V(P 1) = P2" We then write P1 -~ P2" Phase space Let Q be the configuration manifold of a mechanical system. The cotangent bundle T*Q is denoted by P and is called the phase manifold of the system. The traditional coordinate system on P uses position coordinates q = (ql ..... qn) and momentum coordinates p = (p 1 ..... Pn). In a static mechanical system the p j are force coordinates. There is a canonical 1-form 'V'Q on P, the Liouville form, defined by the work done in traversing the stationary paths in Q. The manifold P together with the 2-form o = d'~Q defines a symplectic manifold (P,CO) called the phase space of the system. Examples 2.2 1 The space P --- {(V,S,p,T) ~ ~14 : V > 0, S > 0, p > 0, T > 0} with symplectic form co = dV A dp + dT ^ dS is the phase space for a simple thermodynamic system, in fact a 11

225 perfect gas with V = volume, S = entropy, p = pressure, T = temperature. Here the extensive coordinates V,S correspond to positions and the intensive ones p, T correspond to forces: see Sommerfeld [1964]. 2 Let P be the phase manifold of a mechanical system. Let cp : TP ~ T*P be the diffeomorphism defined by ¢p(v)(u) -- co(v,u), where u,v are vectors in TP such that Xp(U) = Xp(V) for the canonical projection Xp : TP~ P. The pair (TP, dcp*Y'p) defines the phase space for particle (Hamiltonian) dynamics on Q. In local coordinates (q,p,~l,15) on TP we have n d(¢p*~/'p) -- ~ (d~i ^ dqi - dcli A dPi), n = dim Q. i=l See Tulczyjew [1974] for further details. Darboux coordinates Coordinates on P in which co can be written as n co = ~l dpi A dqi are called Darboux coordinates. They always exist locally on (P,co), see Guillemin and Sternberg [1984] p.155. So all symplectic manifolds are locally symplectomorphic, that is (P, co) ~, ([12n, i__~l dPi ^ dqi). n The local Darboux coordinates on P reduce a hypersurface V in P to the local normal form V = {(p,q) : q 1 = 0}, see Arnold [1983]. An analogous fact is true if V is a singular variety, e.g. an ordinary swallowtail variety defined in P, see Arnold [1981]. Lagrangian submanifolds Let (P,co) be a symplectic manifold, dim P --- 2n. A submanifold L of P such that c01L = 0 is said to be isotropic for co. If further dim L = n, its maximum value in this case, then L is called a Lagrangian submanifold of (P,co). The concept of a Lagrangian submanifold is of central importance in symplectic geometry, see Weinstein [19811. Examples 2.3 1 In the phase space of Example 2.2.1 the state equations of a concrete thermodynamical system, say a perfect gas, describe a Lagrangian submanifold of attainable states ~,-1) L -- {(V,S,p,T) ~ P : pV = RT, pV 7 --- ke , R,7,k constants}.

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