16 2. SYMPLECTIC GEOMETRY AND ANALYSIS 2.3. SYMPLECTIC MAPPINGS Suppose next that U, V ⊂ R2n are open sets and κ : U → V is a smooth mapping. We will write κ(x, ξ) = (y, η) = (y(x, ξ),η(x, ξ)). DEFINITION. We call κ a symplectic mapping, or a symplectomorphism, provided (2.3.1) κ∗σ = σ. Here the pull-back κ∗σ of the symplectic product σ is defined by (κ∗σ)(z, w) := σ(κ∗(z),κ∗(w)), κ∗ denoting the push-forward of vectors see Appendix B. NOTATION. We will usually write (2.3.1) in the more suggestive notation (2.3.2) dη ∧ dy = dξ ∧ dx. The ensuing sequence of examples will clarify the meaning of this nota- tion: EXAMPLE 1: Linear symplectic mappings. Suppose κ : R2n → R2n is linear: κ(x, ξ) = A B C D x ξ = (Ax + Bξ, Cx + Dξ) = (y, η), where A, B, C, D are n × n matrices. THEOREM 2.3 (Symplectic matrices). The linear mapping κ is sym- plectic if and only if the matrix K := A B C D satisfies (2.3.3) KT JK = J. In particular the linear mapping (x, ξ) → (ξ, −x) determined by J is symplectic. DEFINITION. We call a 2n × 2n matrix K symplectic if (2.3.3) holds.

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