2.2. SYMPLECTIC STRUCTURE ON
R2n
15
It is convenient to recast σ, employing a very useful matrix:
DEFINITION. We introduce the 2n × 2n matrix
(2.2.2) J :=
0 I
−I 0
.
LEMMA 2.2 (Properties of J and σ).
(i) We have
(2.2.3) σ(z, w) = Jz, w
for all z, w ∈
R2n,
where ·, · now means the standard inner product on
R2n.
(ii) The bilinear form σ is antisymmetric:
(2.2.4) σ(z, w) = −σ(w, z)
and nondegenerate:
(2.2.5) if σ(z, w) = 0 for all w, then z = 0.
(iii) Also
(2.2.6)
J2
= −I,
JT
= −J =
J−1.
We leave the simple proofs to the reader.
We now bring in the terminology of differential forms, reviewed in Ap-
pendix B:
NOTATION. We introduce for x = (x1,...,xn) and ξ = (ξ1,...,ξn) the
1-forms dxj and dξj for j = 1,...,n and then write
(2.2.7) σ = dξ ∧ dx =
n
j=1
dξj ∧ dxj.
Observe also that
(2.2.8) σ = dω for ω := ξdx =
n
j=1
ξjdxj.
Since
d2
= 0, it follows that
(2.2.9) dσ = 0.
