2.3. SYMPLECTIC MAPPINGS 19
Therefore
dy = (A dx)
and
dx = (Edx Fdη) dx = Edx dx + 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
dy = d(−∂yϕ) dy
= [(−∂yϕdy)
2
dy] + [(−∂xyϕdx)
2
dy]
= −(∂xyϕ)dx
2
dy,
since ∂y

is symmetric. Likewise,
dx = d(∂xϕ) dx
= [(∂xϕ
2
dx) dx] + [(∂xyϕ
2
dy) dx]
= −(∂xyϕ)dx
2
dy = dy.
Previous Page Next Page