26 2. SYMPLECTIC GEOMETRY AND ANALYSIS

We readily check that Λκ is a Lagrangian submanifold of

R2n

×

R2n,

with

the symplectic form σ = dη ∧ dy + dξ ∧ dx.

If the map

(x, y, ξ, η) → (y, x)

has a nonvanishing differential on Λκ, we can employ (x, y) as coordinates

in Theorem 2.14:

Λκ = {(x, y, ∂xϕ(x, y),∂yϕ(x, y)) | x, y ∈

Rn}

.

Then (2.5.6) shows that this is equivalent to (2.3.11).

(iii) Another interesting class of generating functions for symplectic maps

appears when

(x, y, ξ, η) → (x, η)

has nonvanishing differential on Λκ. Then

Λκ = {(x, −∂ηϕ(x, η),∂xϕ(x, η),η) | x, η ∈

Rn}

= {(x, ∂ηψ(x, η),∂xψ(x, η), −η) | x, η ∈

Rn}

(2.5.7)

for ψ(x, η) := ϕ(x, −η). This means that

κ(∂ηψ(x, η),η) = (x, ∂xψ(x, η)).

2.6. NOTES

The proof of Theorem 2.12 is from Moser [Mo]; see also Cannas da Silva

[CdS]. A PDE-oriented introduction to symplectic geometry may be found

in H¨ ormander [H3, Chapter 21].

In Greek, the word “symplectic” means “intertwined”. This is consistent

with Example 4, since the generating function ϕ = ϕ(x, y) is a function of a

mixture of half of the original variables (x, ξ) and half of the new variables

(y, η). “Symplectic” can also be interpreted as “complex”, mathematical

usage due to H. Weyl who renamed “line complex group” the “symplectic

group”; see Cannas da Silva [CdS].