2.3. SYMPLECTIC MAPPINGS 19
dη ∧ dy = dη ∧ (A dx)
dξ ∧ dx = (Edx ∧ Fdη) ∧ dx = Edx ∧ dx + dη ∧ F
We would like to construct ξ = ξ(x, η) so that
A = F
and E is symmetric,
the latter condition implying that Edx ∧ dx = 0. To do so, let us define
ξ(x, η) :=
Then clearly F
= A, and E =
EXAMPLE 4: Generating functions. Our next example demonstrates
that we can, locally at least, build a symplectic transformation from a real-
valued generating function.
Suppose ϕ :
→ R, ϕ = ϕ(x, y), is smooth. Assume also that
(2.3.10) ξ = ∂xϕ, η = −∂yϕ
and observe that the Implicit Function Theorem implies (y, η) is a smooth
function of (x, ξ) near (x0,∂xϕ(x0,y0)).
THEOREM 2.7 (Generating functions and symplectic maps). The
mapping κ implicitly defined by
(2.3.11) (x, ∂xϕ(x, y)) → (y, −∂yϕ(x, y))
is a symplectomorphism near (x0,ξ0).
A simple example is ϕ(x, y) = x, y , which generates the linear sym-
plectic mapping represented by the matrix J.
Proof. We compute
dη ∧ dy = d(−∂yϕ) ∧ dy
∧ dy] + [(−∂xyϕdx)
is symmetric. Likewise,
dξ ∧ dx = d(∂xϕ) ∧ dx
dx) ∧ dx] + [(∂xyϕ
dy) ∧ dx]
∧ dy = dη ∧ dy.