2.4. HAMILTONIAN VECTOR FIELDS 21
Proof. A direct calculation verifies assertion (i), and we observe that
H{f,g}h = [Hf , Hg]h
is a rewriting of (2.4.6).
REMARK: Another derivation of Jacobi’s identity. An alternative
proof of (2.4.6) follows, this illustrating the essential property that = 0.
Theorem B.1 provides the identity
0 = dσ(Hf , Hg,Hh)
= Hf σ(Hg,Hh) + Hgσ(Hh,Hf ) + Hhσ(Hf , Hg)
σ([Hf , Hg],Hh) σ([Hg,Hh],Hf ) σ([Hh,Hf ],Hg).
(2.4.8)
Now (2.4.4) implies
Hf σ(Hg,Hh) = {f, {g, h}}
and
σ([Hf , Hg],Hh) = [Hf , Hg]h = Hf Hgh HgHf h
= {f, {g, h}} {g, {f, h}}.
Similar identities hold for other terms. Substituting into (2.4.8) gives Ja-
cobi’s identity.
THEOREM 2.10 (Jacobi’s Theorem). If κ is a symplectomorphism,
then
(2.4.9) Hf = κ∗(Hκ∗f ).
In other words, the pull-back of a Hamiltonian vector field generated by
f,
(2.4.10)
κ∗Hf
:=
(κ−1)∗Hf
,
is the Hamiltonian vector field generated by the pull-back of f.
Proof. Using the notation of (2.4.10),
κ∗(Hf
) σ =
κ∗(Hf
)
κ∗σ
=
κ∗(Hf
σ)
=
−κ∗(df)
=
−d(κ∗f)
= Hκ∗f σ.
Since σ is nondegenerate, (2.4.9) follows.
EXAMPLE. Define κ = J, so that κ(x, ξ) = (ξ, −x). We have
κ∗f(x,
ξ) =
f(ξ, −x), and therefore
Hκ∗f = ∂xf(ξ, −x),∂x + ∂ξf(ξ, −x),∂ξ .
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