20 2. SYMPLECTIC GEOMETRY AND ANALYSIS

Section 2.5 will generalize this example and provide more geometric insight.

2.4. HAMILTONIAN VECTOR FIELDS

We will later be very concerned with dynamics compatible with the symplec-

tic structure introduced earlier. These dynamics are generated by Hamil-

tonian vector fields:

DEFINITION. Given f ∈ C∞(R2n), we define the corresponding Hamil-

tonian vector field, Hf , by requiring

(2.4.1) σ(z, Hf ) = df(z) for all z = (x, ξ).

So

(2.4.2) Hf = ∂ξf, ∂x − ∂xf, ∂ξ =

n

j=1

(fξj ∂xj − fxj ∂ξj ).

Another way of writing Hf uses the contraction defined in Appendix

B:

LEMMA 2.8 (Differentials and Hamiltonian vector fields). We have

(2.4.3) df = −(Hf σ).

Proof. This follows directly from the definition, as we can calculate for each

z that

(Hf σ)(z) = σ(Hf , z) = −σ(z, Hf ) = −df(z).

DEFINITION. If f, g ∈

C∞(R2n),

we define their Poisson bracket

(2.4.4) {f, g} := Hf g = σ(∂f, ∂g).

That is,

(2.4.5) {f, g} = ∂ξf, ∂xg − ∂xf, ∂ξg =

n

j=1

(fξj gxj − fxj gξj ).

LEMMA 2.9 (Brackets, commutators).

(i) We have Jacobi’s identity

(2.4.6) {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0

for all functions f, g, h ∈

C∞(R2n).

(ii) Furthermore,

(2.4.7) H{f,g} = [Hf , Hg].