. 17
( 32 .)


Ta·ξ (G · ξ) ’’ Tb·ξ (G · ξ)
All this shows that there is a well-de¬ned, non-degenerate 2-form „¦ξ on G·ξ which satis¬es
φ— („¦ξ ) = dωξ . Since φ is a smooth submersion, the equation

φ— (d„¦ξ ) = d(dωξ ) = 0

implies that d„¦ξ = 0, as promised.
Note that a consequence of Proposition 3 is that all of the coadjoint orbits are actually
even dimensional. As we shall see when we take up the subject of reduction, the coadjoint
orbits are particularly interesting symplectic manifolds.
Examples: Let G = O(n), with Lie algebra so(n), the space of skew-symmetric n-by-n
matrices. Now there is an O(n)-equivariant positive de¬nite pairing of so(n) with itself ,
given by
x, y = ’tr(xy).
Thus, we can identify so(n)— with so(n) by this pairing. The reader can check that, in this
case, the coadjoint action is isomorphic to the adjoint action

Ad(a)(x) = axa’1 .

If ξ is the rank 2 matrix « 0 ’1 
ξ=1 0 ,
0 0
then it is easy to check that the stabilizer Gξ is just the set of matrices of the form


where a ∈ SO(2) and A ∈ O(n ’ 2). The quotient O(n)/ SO(2) — O(n ’ 2) thus has a
symplectic structure. It is not di¬cult to see that this homogeneous space can be identi¬ed
with the space of oriented 2-planes in En .
As another example, if n = 2m, then Jm lies in so(2m), and its stabilizer is U(m) ‚
SO(2m). It follows that the quotient space SO(2m)/U(m), which is identi¬able as the set
of orthogonal complex structures on E2m , is a symplectic space.
Finally, if G = U(n), then, again, we can identify u(n)— with u(n) via the U(n)-
invariant, positive de¬nite pairing

x, y = ’Re tr(xy) .

L.5.8 87
Again, under this identi¬cation, the coadjoint action agrees with the adjoint action. For
0 < p < n, the stabilizer of the element

iIp 0
ξp =

is easily seen to be U(p) — U(n ’ p). The orbit of ξp is identi¬able with the space Grp (Cn ),
i.e., the Grassmannian of (complex) p-planes in Cn , and, by Proposition 3, carries a canon-
ical, U(n)-invariant symplectic structure.

Darboux™ Theorem. There is a manifold analogue of Proposition 1 which says that
symplectic manifolds of a given dimension are all locally “isomorphic”. This fundamental
result is known as Darboux™ Theorem. I will give the classical proof (due to Darboux) here,
deferring the more modern proof (due to Weinstein) to the next section.

Theorem 1: (Darboux™ Theorem) If „¦ is a closed 2-form on a manifold M 2n which
satis¬es the condition that „¦n be nowhere vanishing, then for every p ∈ M, there is a
neighborhood U of p and a coordinate system x1 , x2 , . . . , xn , y 1 , y 2 , . . . , y n on U so that

„¦|U = dx1 § dy 1 + dx2 § dy 2 + · · · + dxn § dy n .

Proof: We will proceed by induction on n. Assume that we know the theorem for
n’1 ≥ 0. We will prove it for n. Fix p, and let y 1 be a smooth function on M for which
dy 1 does not vanish at p. Now let X be the unique (smooth) vector ¬eld which satis¬es

X „¦ = dy 1 .

This vector ¬eld does not vanish at p, so there is a function x1 on a neighborhood U of p
which satis¬es X(x1 ) = 1. Now let Y be the vector ¬eld on U which satis¬es

„¦ = ’dx1 .

Since d„¦ = 0, the Cartan formula, now gives

LX „¦ = LY „¦ = 0.

We now compute

„¦) ’ Y
[X, Y ] „¦ = LX Y „¦ = LX (Y (LX „¦)
= LX (’dx1 ) = ’d X(x1 ) = ’d(1) = 0.

Since „¦ has maximal rank, this implies [X, Y ] = 0. By the simultaneous ¬‚ow-box theorem,
it follows that there exist local coordinates x1 , y 1 , z 1 , z 2 , . . . , z 2n’2 on some neighborhood
U1 ‚ U of p so that
‚ ‚
X= and Y= .
‚y 1

L.5.9 88
Now consider the form „¦ = „¦ ’ dx1 §dy 1 . Clearly d„¦ = 0. Moreover,

X „¦ = LX „¦ = Y „¦ = LY „¦ = 0.

It follows that „¦ can be expressed as a 2-form in the variables z 1 , z 2 , . . . , z 2n’2 alone.
Hence, in particular, („¦ )n+1 ≡ 0. On the other hand, by the binomial theorem, then

0 = „¦n = n dx1 § dy 1 § („¦ )n’1 .

It follows that „¦ may be regarded as a closed 2-form of maximal half-rank n’1 on an
open set in R2n’2 . Now apply the inductive hypothesis to „¦ .

Darboux™ Theorem has a generalization which covers the case of closed 2-forms of
constant (though not necessarily maximal) rank. It is the analogue for manifolds of the
symplectic reduction of a vector space.

Theorem 2: (Darboux™ Reduction Theorem) Suppose that „¦ is a closed 2-form of constant
half-rank n on a manifold M 2n+k . Then the “null bundle”

N„¦ = v ∈ T M | „¦(v, w) = 0 for all w ∈ Tπ(v) M

is integrable and of constant rank k. Moreover, any point of M has a neighborhood U on
which there exist local coordinates x1 , . . . , xn , y 1 , . . . , y n , z 1 , . . . z k in which

„¦|U = dx1 § dy 1 + dx2 § dy 2 + · · · + dxn § dy n .

Proof: Note that a vector ¬eld X on M is a section of N„¦ if and only if X „¦ = 0. In
particular, since „¦ is closed, the Cartan formula implies that LX „¦ = 0 for all such X.
If X and Y are two sections of N„¦ , then

„¦) ’ Y (LX „¦) = 0 ’ 0 = 0,
[X, Y ] „¦ = LX (Y

so it follows that [X, Y ] is a section of N„¦ as well. Thus, N„¦ is integrable.
Now apply the Frobenius Theorem. For any point p ∈ M, there exists a neighborhood
U on which there exist local coordinates z 1 . . . , z 2n+k so that N„¦ restricted to U is spanned
by the vector ¬elds Zi = ‚/‚z i for 1 ¤ i ¤ k. Since Zi „¦ = LZi „¦ = 0 for 1 ¤ i ¤ k,
it follows that „¦ can be expressed on U in terms of the variables z k+1 , . . . , z 2n+k alone.
In particular, „¦ restricted to U may be regarded as a non-degenerate closed 2-form on an
open set in R2n . The stated result now follows from Darboux™ Theorem.

L.5.10 89
Symplectic and Hamiltonian vector ¬elds.

We now want to examine some of the special vector ¬elds which are de¬ned on sym-
plectic manifolds. Let M 2n be manifold and let „¦ be a symplectic form on M. Let
Sp(„¦) ‚ Diff(M) denote the subgroup of symplectomorphisms of (M, „¦). We would like
to follow Lie in regarding Sp(„¦) as an “in¬nite dimensional Lie group”. In that case,
the Lie algebra of Sp(„¦) should be the space of vector ¬elds whose ¬‚ows preserve „¦. Of
course, „¦ will be invariant under the ¬‚ow of a vector ¬eld X if and only if LX „¦ = 0. This
motivates the following de¬nition:
De¬nition 4: A vector ¬eld X on M is said to be symplectic if LX „¦ = 0. The space of
symplectic vector ¬elds on M will be denoted sp(„¦).
It turns out that there is a very simple characterization of the symplectic vector ¬elds
on M: Since d„¦ = 0, it follows that for any vector ¬eld X on M,
LX „¦ = d(X „¦).
Thus, X is a symplectic vector ¬eld if and only if X „¦ is closed.
Now, since „¦ is non-degenerate, for any vector ¬eld X on M, the 1-form (X) = ’X „¦
vanishes only where X does. Since T M and T — M have the same rank, it follows that the
mapping : X(M) ’ A1 (M) is an isomorphism of C ∞ (M)-modules. In particular, has
an inverse, : A1 (M) ’ X(M).
With this notation, we can write sp(„¦) = Z 1 (M) where Z 1 (M) denotes the vector
space of closed 1-forms on M. Now, Z 1 (M) contains, as a subspace, B 1 (M) = d C ∞ (M) ,
the space of exact 1-forms on M. This subspace is of particular interest; we encountered
it already in Lecture 4.
De¬nition 5: For each f ∈ C ∞(M), the vector ¬eld Xf = (df) is called the Hamiltonian
vector ¬eld associated to f. The set of all Hamiltonian vector ¬elds on M is denoted h(„¦).
Thus, by de¬nition, h(„¦) = B 1(M) . For this reason, Hamiltonian vector ¬elds are
often called exact. Note that a Hamiltonian vector ¬eld is one whose equations, written in
symplectic coordinates, represent an ODE in Hamiltonian form.
The following formula shows that, not only is sp(„¦) a Lie algebra of vector ¬elds, but
that h(„¦) is an ideal in sp(„¦), i.e., that [sp(„¦), sp(„¦)] ‚ h(„¦).

For X, Y ∈ sp(„¦), we have
Proposition 4:
[X, Y ] = X„¦(X,Y ) .
In particular, [Xf , Xg ] = X{f,g} where, by de¬nition, {f, g} = „¦(Xf , Xg ).

Proof: We use the fact that, for any vector ¬eld X, the operator LX is a derivation with
respect to any natural pairing between tensors on M:
„¦ = LX Y „¦ ’ Y
[X, Y ] „¦ = LX Y LX „¦
=d X Y „¦ + X d (Y „¦) + 0 = d „¦(Y, X) + 0
= ’d „¦(X, Y ) = X„¦(X,Y ) „¦.
This proves our ¬rst equation. The remaining equation follows immediately.

L.5.11 90
The de¬nition {f, g} = „¦(Xf , Xg ) is an important one. The bracket (f, g) ’ {f, g} is
called the Poisson bracket of the functions f and g. Proposition 4 implies that the Poisson
bracket gives the functions on M the structure of a Lie algebra. The Poisson bracket is
slightly more subtle than the pairing (Xf , Xg ) ’ X{f,g} since the mapping f ’ Xf has a
non-trivial kernel, namely, the locally constant functions.
Thus, if M is connected, then we get an exact sequence of Lie algebras

0 ’’ R ’’ C ∞ (M) ’’ h(„¦) ’’ 0

which is not, in general, split (see the Exercises). Since {1, f} = 0 for all functions f on
M, it follows that the Poisson bracket on C ∞ (M) makes it into a central extension of
the algebra of Hamiltonian vector ¬elds. The geometry of this central extension plays an
important role in quantization theories on symplectic manifolds (see [GS 2] or [We]).
Also of great interest is the exact sequence

0 ’’ h(„¦) ’’ sp(„¦) ’’ HdR (M, R) ’’ 0,

where the right hand arrow is just the map described by X ’ [X „¦]. Since the bracket of
two elements in sp(„¦) lies in h(„¦), it follows that this linear map is actually a Lie algebra
homomorphism when HdR (M, R) is given the abelian Lie algebra structure. This sequence

also may or may not split (see the Exercises), and the properties of this extension have a
great deal to do with the study of groups of symplectomorphisms of M. See the Exercises
for further developments.

I now want to make some remarks about the meaning of the Poisson bracket and its
De¬nition 5: Let (M, „¦) be a symplectic manifold. Two functions f and g are said to
be in involution (with respect to „¦) if they satisfy the condition {f, g} = 0.
Note that, since {f, g} = dg(Xf ) = ’df(Xg ), it follows that two functions f and g are
in involution if and only if each is constant on the integral curves of the other™s Hamiltonian
vector ¬eld.
Now, if one is trying to describe the integral curves of a Hamiltonian vector ¬eld,
Xf , the more independent functions on M that one can ¬nd which are constant on the
integral curves of Xf , the more accurately one can describe those integral curves. If one
were able ¬nd, in addition to f itself, 2n’2 additional independent functions on M which
are constant on the integral curves of Xf , then one could describe the integral curves of
Xf implicitly by setting those functions equal to a constant.
It turns out, however, that this is too much to hope for in general. It can happen that
a Hamiltonian vector ¬eld Xf has no functions in involution with it except for functions
of the form F (f).

L.5.12 91
Nevertheless, in many cases which arise in practice, we can ¬nd several functions in
involution with a given function f = f1 and, moreover, in involution with each other.
In case one can ¬nd n’1 such independent functions, f2 , . . . , fn , we have the following
theorem of Liouville which says that the remaining n’1 required functions can be found
(at least locally) by quadrature alone. In the classical language, a vector ¬eld Xf for which
such functions are known is said to be “completely integrable by quadratures”, or, more
simply as “completely integrable”.

Theorem 3: Let f 1 , f 2 , . . . , f n be n functions in involution on a symplectic manifold
(M 2n , „¦). Suppose that the functions f i are independent in the sense that the di¬erentials
df 1 , . . . , df n are linearly independent at every point of M. Then each point of M has an
open neighborhood U on which there are functions a1 , . . . , an on U so that

„¦ = df 1 § da1 + · · · + df n § dan .

Moreover, the functions ai can be found by “¬nite” operations and quadrature.

Proof: By hypothesis, the forms df 1 , . . . , df n are linearly independent at every point of
M, so it follows that the Hamiltonian vector ¬elds Xf 1 , . . . , Xf n are also linearly inde-
pendent at every point of M. Also by hypothesis, the functions f i are in involution, so it
follows that df i (Xf j ) = 0 for all i and j.
The vector ¬elds Xf i are linearly independent on M, so by “¬nite” operations, we
¯ ¯
can construct 1-forms β1 , . . . , βn which satisfy the conditions

βi (Xf j ) = δij (Kronecker delta).

Any other set of forms βi which satisfy these conditions are given by expressions:

βi = βi + gij df j .


. 17
( 32 .)