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) = Hfσ(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 Hfσ(Hg,Hh) = {f, {g, h}} and σ([Hf,Hg],Hh) = [Hf,Hg]h = HfHgh HgHfh = {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∗(H f ) σ = κ∗(H f ) κ∗σ = κ∗(H f 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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