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.
