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 dσ = 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 .