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].
