18 2. SYMPLECTIC GEOMETRY AND ANALYSIS
EXAMPLE 2: Nonlinear symplectic mappings. Assume next that
κ(x, ξ) = (y, η)
for smooth functions y = y(x, ξ),η = η(x, ξ). Its linearization is the 2n × 2n
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)
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
EXAMPLE 3: Lifting diffeomorphisms. Let
be a diffeomorphism on
with nondegenerate Jacobian matrix ∂γ = ∂xγ.
We propose to extend γ to a symplectomorphism
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) :=
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,
A := ∂xy, E := ∂xξ, F := ∂ηξ.