16 2. SYMPLECTIC GEOMETRY AND ANALYSIS
2.3. SYMPLECTIC MAPPINGS
Suppose next that U, V ⊂
R2n
are open sets and
κ : U → V
is a smooth mapping. We will write
κ(x, ξ) = (y, η) = (y(x, ξ),η(x, ξ)).
DEFINITION. We call κ a symplectic mapping, or a symplectomorphism,
provided
(2.3.1)
κ∗σ
= σ.
Here the pull-back
κ∗σ
of the symplectic product σ is defined by
(κ∗σ)(z,
w) := σ(κ∗(z),κ∗(w)),
κ∗ denoting the push-forward of vectors; see Appendix B.
NOTATION. We will usually write (2.3.1) in the more suggestive notation
(2.3.2) dη ∧ dy = dξ ∧ dx.
The ensuing sequence of examples will clarify the meaning of this nota-
tion:
EXAMPLE 1: Linear symplectic mappings. Suppose κ :
R2n
→
R2n
is linear:
κ(x, ξ) =
A B
C D
x
ξ
= (Ax + Bξ, Cx + Dξ) = (y, η),
where A, B, C, D are n × n matrices.
THEOREM 2.3 (Symplectic matrices). The linear mapping κ is sym-
plectic if and only if the matrix
K :=
A B
C D
satisfies
(2.3.3)
KT
JK = J.
In particular the linear mapping (x, ξ) → (ξ, −x) determined by J is
symplectic.
DEFINITION. We call a 2n × 2n matrix K symplectic if (2.3.3) holds.
