14 2. SYMPLECTIC GEOMETRY AND ANALYSIS
where ˙ = ∂t and w = w(t) = w(t, z), the latter notation used when we
wish to display the dependence of the solution on the initial condition. We
assume that the solution of the flow (2.1.1) exists for each z and is unique
for all times t R.
NOTATION. We define
ϕtz := w(t, z)
and will often also write
(2.1.2) ϕt =: exp(tV ).
We call {ϕt}t∈R the flow map or the exponential map generated by V .
The following are standard assertions from the theory of ordinary dif-
ferential equations:
LEMMA 2.1 (Properties of flow map).
(i) ϕ0z = z for all z RN .
(ii) ϕt+s = ϕtϕs for all s, t R.
(iii) For each time t R, the mapping ϕt :
RN

RN
is a diffeomor-
phism, with
(ϕt)−1
= ϕ−t.
2.2. SYMPLECTIC STRUCTURE ON
R2n
We henceforth specialize to the even-dimensional space
RN
=
R2n
=
Rn
×
Rn.
NOTATION. We refine our previous notation and henceforth denote an
element of
R2n
as
z = (x, ξ)
and interpret x
Rn
as denoting position, ξ
Rn
as momentum. We will
likewise write
w = (y, η)
for another typical point of
R2n.
We let ·, · denote the usual inner product on
Rn,
and then define this
new pairing on
R2n:
DEFINITION. Given z = (x, ξ), w = (y, η) in
R2n,
define their symplectic
product
(2.2.1) σ(z, w) := ξ, y x, η .
Previous Page Next Page