d
dt φ∗
t{F, G}(x)def. of pullback
=d
dt {F, G}(φt(x))
chain rule
=d{F, G}d
dt φt(x)
def. of φt
=d{F, G}vH(φt(x))
def. of vH
=vH{F, G}(φt(x))
Jacobi id.
={H, {F, G}} (φt(x))
= ({{H, F }, G}+{F, {H, G}})φt(x)
On the other hand:
d
dt {φ∗
tF, φ∗
tG}(x)bilin. of {·,·} gives prod. rule
= d
dt φ∗
tF, φ∗
tG+φ∗
tF, d
dt φ∗
tG(x)
={{H, φ∗
tF}, φ∗
tG}+{φ∗
tF, {H, φ∗
tG}} (x)
This agrees with the other side at t= 0.
(Note: H=φ∗
tHsince energy is conserved.)
So we have: d
dt φ∗
t{F, G}(x)
t=0
=d
dt {φ∗
tF, φ∗
tG}(x)
t=0
for all x, so use φs(x)∈X:
d
dt φ∗
t{F, G}φs(x)
t=0
=d
dt {φ∗
tF, φ∗
tG}(φs(x))
t=0
d
dt {F, G}φt+s(x)
t=0
=
d
ds {F, G}φs(x) =
We would like: d
ds {F, G}φs(x) = d
ds {φ∗
sF, φ∗
sG}
Ugh!
So: we have a category of Poisson manifolds and Poisson maps and time evolution for any Hamilto-
nian is a Poisson map.
Definition 2 Acategory consists of a collection of objects and for any pair of objects Xand
Ya set of morphisms f:X→Ysuch that given f:X→Yand g:Y→Zwe have a morphism
gf :X→Z, such that:
1. (hg)f = h(gf )
2. each Xhas an identity morphism 1X:X→Xsuch that:
f1X=f
1Xg=g
2
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