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
dy = (Cdx + Ddξ) (Adx + Bdξ)
for
A := ∂xy, B := ∂ξy, C := ∂xη, D := ∂ξη.
Consequently, as in the previous proof, we have dy = 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, = E dx + F dη,
for
A := ∂xy, E := ∂xξ, F := ∂ηξ.
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