18 2. SYMPLECTIC GEOMETRY AND ANALYSIS EXAMPLE 2: Nonlinear symplectic mappings. Assume next that κ : R2n → R2n is nonlinear: κ(x, ξ) = (y, η) for smooth functions y = y(x, ξ),η = η(x, ξ). Its linearization is the 2n × 2n matrix ∂κ = ∂xy ∂ξy ∂xη ∂ξη . THEOREM 2.5 (Symplectic transformations). The mapping κ is sym- plectic if and only if the matrix ∂κ is symplectic at each point. Proof. We have dη ∧ dy = (Cdx + Ddξ) ∧ (Adx + Bdξ) for A := ∂xy, B := ∂ξy, C := ∂xη, D := ∂ξη. Consequently, as in the previous proof, we have dη ∧ dy = dξ ∧ dx if and only if (2.3.4) is valid, which in turn is so if and only if ∂κ is a symplectic matrix. EXAMPLE 3: Lifting diffeomorphisms. Let γ : Rn → Rn be a diffeomorphism on Rn, with nondegenerate Jacobian matrix ∂γ = ∂xγ. We propose to extend γ to a symplectomorphism κ : R2n → R2n having the form (2.3.7) κ(x, ξ) = (γ(x),η(x, ξ)) = (y, η), by “lifting” γ to variables ξ. THEOREM 2.6 (Extending to a symplectic mapping). The trans- formation (2.3.7) is symplectic if (2.3.8) η(x, ξ) := ∂γ(x)−1 T ξ. Proof. As the statement suggests, it will be easier to look for ξ as a function of x and η. We compute dy = A dx, dξ = E dx + F dη, for A := ∂xy, E := ∂xξ, F := ∂ηξ.

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