10 D. MCDUFF, INTRODUCTION TO SYMPLECTIC TOPOLOGYshows that XH is t a ngent to the level sets of H. Thus the flow of <Pf preserves the
function H.
Example 1.2. With H: R^2 ---+ R given by H(x, y) = y and with w = dx /\ dy, we
get
{)
XH = - and ¢f(x, y) = (x + t, y).
oxIf H : R^2 n ---+ R and w = wo, we get
f)H 8 oH 8
xH = L oyi axi - axi oyi ·
t
The solution curves (x i (t), Yi (t)) = <l>t(x(O), y(O)) satisfy Hamilton's equations
oH. oH
Xi = OYi , Yi = - OXi.With H = ~I:: x] + YJ, the orbits of this action are circles. In complex coordinates
Zj = Xj + iyj we have
,;.,H 'Pt ( Z1, ... , Zn ) = ( e - it Z1 , ... , e -it Zn ).
Thus we get a circle action on R^2 n = e n. The function H that generates it is called
the moment map of this action.Exercise 1.3. Often it is useful to consider Hamiltonian functions that depend on
time: viz:H: M x [O, 1] ---+ R, H(p, t) = H 1 (p).
Then one defines XH, as before, and gets a smooth family <Pf of symplectomor-
phisms with <Pt/ =id which at time t are tangent to XH, :
d H Hdt (t (p)) = XH, (t (p)), p EM, t E [0, l].
Such a family is called a Hamiltonian isotopy. Show that the set of all time-1
maps ¢f^1 forms a subgroup of Symp(M). This is called the group of Hamiltonian
symplectomorphisms Ham(M). Its elements are also often called exact symplecto-
morphisms.Linear symplectic geometry
To get a better understanding of what is going on, let's now look at what happens
at a point. As we shall see, linear symplectic geometry contains a surprising amount
of structure. Moreover, most of this structure at a point corresponds very clearly to
nonlinear phenomena. One example of this is Darboux's theorem. We shall see in
a minute that there is only one symplectic structure on a given (finite-dimensional)
vector space, up to isomorphism. Darboux's theorem says that, locally, there is
only one symplectic form on a smooth manifold. In other words, every symplectic
form w on M is locally symplectomorphic to the standard form wo on R^2 n. One
might think that this implies there is no interesting local structure (just as if one
were in the category of smooth manifolds.) But this is false, since, as we shall see,
the standard structure w 0 on R^2 n is itself very interesting.
So let V be a vector space (over R) with a nondegenerate skew bilinear form
w. Thus