2.2. SYMPLECTIC STRUCTURE ON
It is convenient to recast σ, employing a very useful matrix:
DEFINITION. We introduce the 2n × 2n matrix
(2.2.2) J :=
LEMMA 2.2 (Properties of J and σ).
(i) We have
(2.2.3) σ(z, w) = Jz, w
for all z, w ∈
where ·, · now means the standard inner product on
(ii) The bilinear form σ is antisymmetric:
(2.2.4) σ(z, w) = −σ(w, z)
(2.2.5) if σ(z, w) = 0 for all w, then z = 0.
= −J =
We leave the simple proofs to the reader.
We now bring in the terminology of differential forms, reviewed in Ap-
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 =
dξj ∧ dxj.
Observe also that
(2.2.8) σ = dω for ω := ξdx =
= 0, it follows that
(2.2.9) dσ = 0.