2.5. LAGRANGIAN SUBMANIFOLDS 25 with ρ(0) = (x0,ξ0). The Jacobian ∂ρ(0) has full rank and hence has n inde- pendent rows. We choose n such rows and call the corresponding coordinates x Rk and ξ Rn−k. 2. Define p : R2n Rk × Rn−k by p(x, ξ) := (x , ξ ). Our choice of the coordinates (x , ξ ) means that p ρ : V Rk × Rn−k has an invertible Jacobian at 0 Rn. Hence the Implicit Function Theorem implies p γ is invertible in a neighborhood of 0. This means that we can use (x , ξ ) as local coordinates on Λ, and so there exists a neighborhood U R2n of (x0,ξ0) and smooth functions f : Rk × Rn−k Rn−k, g : Rk × Rn−k Rk such that Λ U = {(x , f(x , ξ ),g(x , ξ ),ξ ) | (x , ξ ) Rk × Rn−k} U. 3. Recalling that ω = ξdx, we use Theorem 2.13 to see that for some function ψ = ψ(x , ξ ), ω|Λ = g, dx + ξ , ∂x fdx + ∂ξ fdξ = g + (∂x f)T ξ , dx + (∂ξ f)T ξ , = ∂x ψ, dx + ∂ξ ψ, . That is, ψx = g + (∂x f)T ξ = g + ∂x f, ξ , ψξ = (∂ξ f)T ξ . If we put ϕ(x , ξ ) := ψ(x , ξ ) f(x , ξ ),ξ , then f = −∂ξ ϕ, g = ∂x ϕ. EXAMPLES. (i) The simplest case is k = n, for which Λ U = {(x, ∂ϕ(x)) : x Rn}. Then (2.5.2) reads ω|Λ = = ∂ϕdx. (ii) Theorem 2.14 generalizes Theorem 2.7. To see this, consider the twisted graph of κ: (2.5.6) Λκ = (x, y, ξ, −η) | (x, ξ) = κ(y, η), (y, η) R2n .
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