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.