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 = 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
= 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
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