22 2. SYMPLECTIC GEOMETRY AND ANALYSIS

Then

κ∗Hf

= ∂ξf(ξ, −x),∂ξ − ∂xf(ξ, −x), −∂x = Hκ∗f .

THEOREM 2.11 (Hamiltonian flows as symplectomorphisms). If

f is smooth, then for each time t, the mapping

(x, ξ) → ϕt(x, ξ) = exp(tHf )

is a symplectomorphism.

Proof. According to Cartan’s formula (Theorem B.3), we have

∂t(ϕt

∗σ)

= LHf σ = d(Hf σ) + (Hf dσ).

Since dσ = 0, it follows that

∂t(ϕt

∗σ)

= d(−df) =

−d2f

= 0.

Thus

(ϕt)∗σ

= σ for all times t.

The next result shows that locally all nondegenerate closed 2-forms are

equivalent to the standard symplectic form σ on

R2n.

THEOREM 2.12 (Darboux’s Theorem). Let U be a neighborhood of

(x0,ξ0) and suppose η is a nondegenerate 2-form defined on U, satisfying

dη = 0.

Then near (x0,ξ0) there exists a diffeomorphism κ such that

(2.4.11)

κ∗η

= σ.

INTERPRETATION. A symplectic structure on

R2n

is determined by

a choice of a nondegenerate, closed 2-form η. Darboux’s Theorem states

that all symplectic structures are identical locally, in the sense that all are

equivalent to that given by σ. This is a dramatic contrast to Riemannian

geometry: there are no local invariants in symplectic geometry.

Proof. 1. Let us assume (x0,ξ0) = (0, 0). We first need a linear mapping L

so that

L∗η(0,

0) = σ(0, 0).

This means that we find a basis {ek,fk}k=1

n

of

R2n

such that

η(fl,ek) = δkl, η(ek,el) = 0, η(fk,fl) = 0

for all 1 ≤ k, l ≤ n. Then if u =

∑n

i=1

xiei + ξifi, v =

∑n

j=1

yjej + ηjfj, we