22 2. SYMPLECTIC GEOMETRY AND ANALYSIS Then κ∗Hf = ∂ξf(ξ, −x),∂ξ− ∂xf(ξ, −x), −∂x = Hκ∗f. THEOREM 2.11 (Hamiltonian flows as symplectomorphisms). If f is smooth, then for each time t, the mapping (x, ξ) → ϕt(x, ξ) = exp(tHf) is a symplectomorphism. Proof. According to Cartan’s formula (Theorem B.3), we have ∂t(ϕt ∗ σ) = LH f σ = d(Hf σ) + (Hf dσ). Since dσ = 0, it follows that ∂t(ϕt ∗ σ) = d(−df) = −d2f = 0. Thus (ϕt)∗σ = σ for all times t. The next result shows that locally all nondegenerate closed 2-forms are equivalent to the standard symplectic form σ on R2n. THEOREM 2.12 (Darboux’s Theorem). Let U be a neighborhood of (x0,ξ0) and suppose η is a nondegenerate 2-form defined on U, satisfying dη = 0. Then near (x0,ξ0) there exists a diffeomorphism κ such that (2.4.11) κ∗η = σ. INTERPRETATION. A symplectic structure on R2n is determined by a choice of a nondegenerate, closed 2-form η. Darboux’s Theorem states that all symplectic structures are identical locally, in the sense that all are equivalent to that given by σ. This is a dramatic contrast to Riemannian geometry: there are no local invariants in symplectic geometry. Proof. 1. Let us assume (x0,ξ0) = (0, 0). We first need a linear mapping L so that L∗η(0, 0) = σ(0, 0). This means that we find a basis {ek,fk}k=1 n of R2n such that η(fl,ek) = δkl, η(ek,el) = 0, η(fk,fl) = 0 for all 1 ≤ k, l ≤ n. Then if u = ∑ n i=1 xiei + ξifi, v = ∑ n j=1 yjej + ηjfj, we

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