26 2. SYMPLECTIC GEOMETRY AND ANALYSIS We readily check that Λκ is a Lagrangian submanifold of R2n × R2n, with the symplectic form σ = dη ∧ dy + dξ ∧ dx. If the map (x, y, ξ, η) → (y, x) has a nonvanishing differential on Λκ, we can employ (x, y) as coordinates in Theorem 2.14: Λκ = {(x, y, ∂xϕ(x, y),∂yϕ(x, y)) | x, y ∈ Rn} . Then (2.5.6) shows that this is equivalent to (2.3.11). (iii) Another interesting class of generating functions for symplectic maps appears when (x, y, ξ, η) → (x, η) has nonvanishing differential on Λκ. Then Λκ = {(x, −∂ηϕ(x, η),∂xϕ(x, η),η) | x, η ∈ Rn} = {(x, ∂ηψ(x, η),∂xψ(x, η), −η) | x, η ∈ Rn} (2.5.7) for ψ(x, η) := ϕ(x, −η). This means that κ(∂ηψ(x, η),η) = (x, ∂xψ(x, η)). 2.6. NOTES The proof of Theorem 2.12 is from Moser [Mo] see also Cannas da Silva [CdS]. A PDE-oriented introduction to symplectic geometry may be found in H¨ ormander [H3, Chapter 21]. In Greek, the word “symplectic” means “intertwined”. This is consistent with Example 4, since the generating function ϕ = ϕ(x, y) is a function of a mixture of half of the original variables (x, ξ) and half of the new variables (y, η). “Symplectic” can also be interpreted as “complex”, mathematical usage due to H. Weyl who renamed “line complex group” the “symplectic group” see Cannas da Silva [CdS].
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