18 2. SYMPLECTIC GEOMETRY AND ANALYSIS

EXAMPLE 2: Nonlinear symplectic mappings. Assume next that

κ :

R2n

→

R2n

is nonlinear:

κ(x, ξ) = (y, η)

for smooth functions y = y(x, ξ),η = η(x, ξ). Its linearization is the 2n × 2n

matrix

∂κ =

∂xy ∂ξy

∂xη ∂ξη

.

THEOREM 2.5 (Symplectic transformations). The mapping κ is sym-

plectic if and only if the matrix ∂κ is symplectic at each point.

Proof. We have

dη ∧ dy = (Cdx + Ddξ) ∧ (Adx + Bdξ)

for

A := ∂xy, B := ∂ξy, C := ∂xη, D := ∂ξη.

Consequently, as in the previous proof, we have dη ∧ dy = dξ ∧ dx if and

only if (2.3.4) is valid, which in turn is so if and only if ∂κ is a symplectic

matrix.

EXAMPLE 3: Lifting diffeomorphisms. Let

γ :

Rn

→

Rn

be a diffeomorphism on

Rn,

with nondegenerate Jacobian matrix ∂γ = ∂xγ.

We propose to extend γ to a symplectomorphism

κ :

R2n

→

R2n

having the form

(2.3.7) κ(x, ξ) = (γ(x),η(x, ξ)) = (y, η),

by “lifting” γ to variables ξ.

THEOREM 2.6 (Extending to a symplectic mapping). The trans-

formation (2.3.7) is symplectic if

(2.3.8) η(x, ξ) :=

∂γ(x)−1

T

ξ.

Proof. As the statement suggests, it will be easier to look for ξ as a function

of x and η. We compute

dy = A dx, dξ = E dx + F dη,

for

A := ∂xy, E := ∂xξ, F := ∂ηξ.