2.3. SYMPLECTIC MAPPINGS 19 Therefore dη ∧ dy = dη ∧ (A dx) and dξ ∧ dx = (Edx ∧ Fdη) ∧ dx = Edx ∧ dx + dη ∧ F T dx. We would like to construct ξ = ξ(x, η) so that A = F T and E is symmetric, the latter condition implying that Edx ∧ dx = 0. To do so, let us define ξ(x, η) := (∂γ)T η. Then clearly F T = A, and E = ET = ∂2γ, as required. EXAMPLE 4: Generating functions. Our next example demonstrates that we can, locally at least, build a symplectic transformation from a real- valued generating function. Suppose ϕ : Rn × Rn → R, ϕ = ϕ(x, y), is smooth. Assume also that (2.3.9) det(∂2 xy ϕ(x0,y0)) = 0. Define (2.3.10) ξ = ∂xϕ, η = −∂yϕ and observe that the Implicit Function Theorem implies (y, η) is a smooth function of (x, ξ) near (x0,∂xϕ(x0,y0)). THEOREM 2.7 (Generating functions and symplectic maps). The mapping κ implicitly defined by (2.3.11) (x, ∂xϕ(x, y)) → (y, −∂yϕ(x, y)) is a symplectomorphism near (x0,ξ0). A simple example is ϕ(x, y) = x, y , which generates the linear sym- plectic mapping represented by the matrix J. Proof. We compute dη ∧ dy = d(−∂yϕ) ∧ dy = [(−∂2ϕdy) y ∧ dy] + [(−∂2 xy ϕdx) ∧ dy] = −(∂xyϕ)dx 2 ∧ dy, since ∂2ϕ y is symmetric. Likewise, dξ ∧ dx = d(∂xϕ) ∧ dx = [(∂2ϕ x dx) ∧ dx] + [(∂2 xy ϕ dy) ∧ dx] = −(∂2 xy ϕ)dx ∧ dy = dη ∧ dy.

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