14 2. SYMPLECTIC GEOMETRY AND ANALYSIS where ˙ = ∂t and w = w(t) = w(t, z), the latter notation used when we wish to display the dependence of the solution on the initial condition. We assume that the solution of the flow (2.1.1) exists for each z and is unique for all times t ∈ R. NOTATION. We define ϕtz := w(t, z) and will often also write (2.1.2) ϕt =: exp(tV ). We call {ϕt}t∈R the flow map or the exponential map generated by V . The following are standard assertions from the theory of ordinary dif- ferential equations: LEMMA 2.1 (Properties of flow map). (i) ϕ0z = z for all z ∈ RN. (ii) ϕt+s = ϕtϕs for all s, t ∈ R. (iii) For each time t ∈ R, the mapping ϕt : RN → RN is a diffeomor- phism, with (ϕt)−1 = ϕ−t. 2.2. SYMPLECTIC STRUCTURE ON R2n We henceforth specialize to the even-dimensional space RN = R2n = Rn × Rn. NOTATION. We refine our previous notation and henceforth denote an element of R2n as z = (x, ξ) and interpret x ∈ Rn as denoting position, ξ ∈ Rn as momentum. We will likewise write w = (y, η) for another typical point of R2n. We let ·, · denote the usual inner product on Rn, and then define this new pairing on R2n: DEFINITION. Given z = (x, ξ), w = (y, η) in R2n, define their symplectic product (2.2.1) σ(z, w) := ξ, y− x, η .

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