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.