2.3. SYMPLECTIC MAPPINGS 19

Therefore

dη ∧ dy = dη ∧ (A dx)

and

dξ ∧ dx = (Edx ∧ Fdη) ∧ dx = Edx ∧ dx + dη ∧ F

T

dx.

We would like to construct ξ = ξ(x, η) so that

A = F

T

and E is symmetric,

the latter condition implying that Edx ∧ dx = 0. To do so, let us define

ξ(x, η) :=

(∂γ)T

η.

Then clearly F

T

= A, and E =

ET

=

∂2γ,

as required.

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 :

Rn

×

Rn

→ R, ϕ = ϕ(x, y), is smooth. Assume also that

(2.3.9) det(∂xyϕ(x0,y0))

2

= 0.

Define

(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

= [(−∂yϕdy)

2

∧ dy] + [(−∂xyϕdx)

2

∧ dy]

= −(∂xyϕ)dx

2

∧ dy,

since ∂y

2ϕ

is symmetric. Likewise,

dξ ∧ dx = d(∂xϕ) ∧ dx

= [(∂xϕ

2

dx) ∧ dx] + [(∂xyϕ

2

dy) ∧ dx]

= −(∂xyϕ)dx

2

∧ dy = dη ∧ dy.