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 .