(using the notationcq = (cq 1 ,...,cqd), cp = (cp 1 ,...,cpd)). We will often
denote these by ((
cq
cp
)
,c)
This Lie bracket onh 2 d+1is given by
[((
cq
cp)
,c)
,
((
c′q
c′p)
,c′)]
=
((
0
0
)
,Ω
((
cq
cp)
,
(
c′q
c′p)))
(14.3)
which depends just on the antisymmetric bilinear form
Ω
((
cq
cp)
,
(
c′q
c′p))
=cq·c′p−cp·c′q (14.4)14.3 Symplectic geometry
We saw in chapter 4 that given a basisejof a vector spaceV, a dual basise∗j
ofV∗is given by takinge∗j=vj, wherevjare the coordinate functions. If one
instead is initially given the coordinate functionsvj, a dual basis ofV= (V∗)∗
can be constructed by taking as basis vectors the first-order linear differential
operators given by differentiation with respect to thevj, in other words by
taking
ej=∂
∂vjElements ofV are then identified with linear combinations of these operators.
In effect, one is identifying vectorsvwith the directional derivative along the
vector
v↔v·∇
We also saw in chapter 4 that an inner product (·,·) onV provides an
isomorphism ofV andV∗by
v∈V↔lv(·) = (v,·)∈V∗ (14.5)Such an inner product is the fundamental structure in Euclidean geometry,
giving a notion of length of a vector and angle between two vectors, as well
as a group, the orthogonal group of linear transformations preserving the inner
product. It is a symmetric, non-degenerate bilinear form onV.
A phase spaceM does not usually come with a choice of inner product.
Instead, we have seen that the Poisson bracket gives us not a symmetric bi-
linear form, but an antisymmetric bilinear form Ω, defined on the dual space
M. We will define an analog of an inner product, with symmetry replaced by
antisymmetry:
Definition(Symplectic form).A symplectic formωon a vector spaceV is a
bilinear map
ω:V×V→R
such that