2.3. SYMPLECTIC MAPPINGS 17

Proof. Let us compute

dη ∧ dy = (Cdx + Ddξ) ∧ (Adx + Bdξ)

=

AT

Cdx ∧ dx +

BT

Ddξ ∧ dξ +

(AT

D −

CT

B)dξ ∧ dx

= dξ ∧ dx

if and only if

(2.3.4)

AT

C and

BT

D are symmetric,

AT

D −

CT

B = I.

Furthermore,

KT

JK =

AT CT

BT DT

O I

−I O

A B

C D

=

AT

C −

CT

A

AT

D −

CT

B

BT C − DT A BT D − DT B

= J

if and only if (2.3.4) holds.

We record some useful observations:

THEOREM 2.4 (More on symplectic matrices).

(i) The product of two symplectic matrices is symplectic.

(ii) If K is a symplectic matrix, then

(2.3.5) σ(Kz, Kw) = σ(z, w) (z, w ∈

R2n).

(iii) A matrix K is symplectic if and only if

(2.3.6) K is invertible,

K−1

=

JKT JT

.

(iv) If

AT

J + JA = 0,

then

Kt := exp(tA) is symplectic

for each t ∈ R.

Proof. Assertions (i), (ii), and (iii) follow directly from the definitions and

the fact that

JT

= −J =

J−1.

To prove (iv), write Wt := Kt

T

JKt − J and

compute

∂tWt =

AT

Wt + WtA +

AT

J + JA =

AT

Wt + WtA.

Since W0 = 0, we deduce from uniqueness that Wt = 0 for all t ∈ R.