ńņš. 17 |

id

TaĀ·Ī¾ (G Ā· Ī¾) ā’ā’ TbĀ·Ī¾ (G Ā· Ī¾)

All this shows that there is a well-deļ¬ned, non-degenerate 2-form ā„¦Ī¾ on GĀ·Ī¾ which satisļ¬es

Ļā— (ā„¦Ī¾ ) = dĻĪ¾ . Since Ļ is a smooth submersion, the equation

Ļā— (dā„¦Ī¾ ) = d(dĻĪ¾ ) = 0

implies that dā„¦Ī¾ = 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 deļ¬nite pairing of so(n) with itself ,

given by

x, y = ā’tr(xy).

Thus, we can identify so(n)ā— 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) = axaā’1 .

If Ī¾ is the rank 2 matrix ļ£« 0 ā’1 ļ£¶

0

Ī¾=ļ£1 0 ļ£ø,

0 0

then it is easy to check that the stabilizer GĪ¾ is just the set of matrices of the form

a0

0A

where a ā SO(2) and A ā O(n ā’ 2). The quotient O(n)/ SO(2) Ć— O(n ā’ 2) thus has a

symplectic structure. It is not diļ¬cult to see that this homogeneous space can be identiļ¬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) ā‚

SO(2m). It follows that the quotient space SO(2m)/U(m), which is identiļ¬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)ā— with u(n) via the U(n)-

invariant, positive deļ¬nite pairing

x, y = ā’Re tr(xy) .

L.5.8 87

Again, under this identiļ¬cation, the coadjoint action agrees with the adjoint action. For

0 < p < n, the stabilizer of the element

iIp 0

Ī¾p =

ā’iInā’p

0

is easily seen to be U(p) Ć— U(n ā’ p). The orbit of Ī¾p is identiļ¬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.

Darbouxā™ Theorem. There is a manifold analogue of Proposition 1 which says that

symplectic manifolds of a given dimension are all locally āisomorphicā. This fundamental

result is known as Darbouxā™ 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: (Darbouxā™ Theorem) If ā„¦ is a closed 2-form on a manifold M 2n which

satisļ¬es the condition that ā„¦n be nowhere vanishing, then for every p ā 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

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

Proof: We will proceed by induction on n. Assume that we know the theorem for

nā’1 ā„ 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 ļ¬eld which satisļ¬es

X ā„¦ = dy 1 .

This vector ļ¬eld does not vanish at p, so there is a function x1 on a neighborhood U of p

which satisļ¬es X(x1 ) = 1. Now let Y be the vector ļ¬eld on U which satisļ¬es

ā„¦ = ā’dx1 .

Y

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

LX ā„¦ = LY ā„¦ = 0.

We now compute

ā„¦) ā’ Y

[X, Y ] ā„¦ = LX Y ā„¦ = LX (Y (LX ā„¦)

= LX (ā’dx1 ) = ā’d X(x1 ) = ā’d(1) = 0.

Since ā„¦ has maximal rank, this implies [X, Y ] = 0. By the simultaneous ļ¬‚ow-box theorem,

it follows that there exist local coordinates x1 , y 1 , z 1 , z 2 , . . . , z 2nā’2 on some neighborhood

U1 ā‚ U of p so that

ā‚ ā‚

X= and Y= .

ā‚y 1

ā‚x1

L.5.9 88

Now consider the form ā„¦ = ā„¦ ā’ dx1 ā§dy 1 . Clearly dā„¦ = 0. Moreover,

X ā„¦ = LX ā„¦ = Y ā„¦ = LY ā„¦ = 0.

It follows that ā„¦ can be expressed as a 2-form in the variables z 1 , z 2 , . . . , z 2nā’2 alone.

Hence, in particular, (ā„¦ )n+1 ā” 0. On the other hand, by the binomial theorem, then

0 = ā„¦n = n dx1 ā§ dy 1 ā§ (ā„¦ )nā’1 .

It follows that ā„¦ may be regarded as a closed 2-form of maximal half-rank nā’1 on an

open set in R2nā’2 . Now apply the inductive hypothesis to ā„¦ .

Darbouxā™ 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: (Darbouxā™ Reduction Theorem) Suppose that ā„¦ is a closed 2-form of constant

half-rank n on a manifold M 2n+k . Then the ānull bundleā

Nā„¦ = v ā T M | ā„¦(v, w) = 0 for all w ā TĻ(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

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

Proof: Note that a vector ļ¬eld X on M is a section of Nā„¦ if and only if X ā„¦ = 0. In

particular, since ā„¦ is closed, the Cartan formula implies that LX ā„¦ = 0 for all such X.

If X and Y are two sections of Nā„¦ , then

ā„¦) ā’ Y (LX ā„¦) = 0 ā’ 0 = 0,

[X, Y ] ā„¦ = LX (Y

so it follows that [X, Y ] is a section of Nā„¦ as well. Thus, Nā„¦ is integrable.

Now apply the Frobenius Theorem. For any point p ā M, there exists a neighborhood

U on which there exist local coordinates z 1 . . . , z 2n+k so that Nā„¦ restricted to U is spanned

by the vector ļ¬elds Zi = ā‚/ā‚z i for 1 ā¤ i ā¤ k. Since Zi ā„¦ = LZi ā„¦ = 0 for 1 ā¤ i ā¤ k,

it follows that ā„¦ can be expressed on U in terms of the variables z k+1 , . . . , z 2n+k alone.

In particular, ā„¦ 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 Darbouxā™ Theorem.

L.5.10 89

Symplectic and Hamiltonian vector ļ¬elds.

We now want to examine some of the special vector ļ¬elds which are deļ¬ned on sym-

plectic manifolds. Let M 2n be manifold and let ā„¦ be a symplectic form on M. Let

Sp(ā„¦) ā‚ Diff(M) denote the subgroup of symplectomorphisms of (M, ā„¦). We would like

to follow Lie in regarding Sp(ā„¦) as an āinļ¬nite dimensional Lie groupā. In that case,

the Lie algebra of Sp(ā„¦) should be the space of vector ļ¬elds whose ļ¬‚ows preserve ā„¦. Of

course, ā„¦ will be invariant under the ļ¬‚ow of a vector ļ¬eld X if and only if LX ā„¦ = 0. This

motivates the following deļ¬nition:

Deļ¬nition 4: A vector ļ¬eld X on M is said to be symplectic if LX ā„¦ = 0. The space of

symplectic vector ļ¬elds on M will be denoted sp(ā„¦).

It turns out that there is a very simple characterization of the symplectic vector ļ¬elds

on M: Since dā„¦ = 0, it follows that for any vector ļ¬eld X on M,

LX ā„¦ = d(X ā„¦).

Thus, X is a symplectic vector ļ¬eld if and only if X ā„¦ is closed.

Now, since ā„¦ is non-degenerate, for any vector ļ¬eld X on M, the 1-form (X) = ā’X ā„¦

vanishes only where X does. Since T M and T ā— M have the same rank, it follows that the

mapping : X(M) ā’ A1 (M) is an isomorphism of C ā (M)-modules. In particular, has

an inverse, : A1 (M) ā’ X(M).

With this notation, we can write sp(ā„¦) = 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 ā (M) ,

the space of exact 1-forms on M. This subspace is of particular interest; we encountered

it already in Lecture 4.

Deļ¬nition 5: For each f ā C ā(M), the vector ļ¬eld Xf = (df) is called the Hamiltonian

vector ļ¬eld associated to f. The set of all Hamiltonian vector ļ¬elds on M is denoted h(ā„¦).

Thus, by deļ¬nition, h(ā„¦) = B 1(M) . For this reason, Hamiltonian vector ļ¬elds are

often called exact. Note that a Hamiltonian vector ļ¬eld is one whose equations, written in

symplectic coordinates, represent an ODE in Hamiltonian form.

The following formula shows that, not only is sp(ā„¦) a Lie algebra of vector ļ¬elds, but

that h(ā„¦) is an ideal in sp(ā„¦), i.e., that [sp(ā„¦), sp(ā„¦)] ā‚ h(ā„¦).

For X, Y ā sp(ā„¦), we have

Proposition 4:

[X, Y ] = Xā„¦(X,Y ) .

In particular, [Xf , Xg ] = X{f,g} where, by deļ¬nition, {f, g} = ā„¦(Xf , Xg ).

Proof: We use the fact that, for any vector ļ¬eld X, the operator LX is a derivation with

respect to any natural pairing between tensors on M:

ā„¦ = LX Y ā„¦ ā’ Y

[X, Y ] ā„¦ = LX Y LX ā„¦

=d X Y ā„¦ + X d (Y ā„¦) + 0 = d ā„¦(Y, X) + 0

= ā’d ā„¦(X, Y ) = Xā„¦(X,Y ) ā„¦.

This proves our ļ¬rst equation. The remaining equation follows immediately.

L.5.11 90

The deļ¬nition {f, g} = ā„¦(Xf , Xg ) is an important one. The bracket (f, g) ā’ {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 ) ā’ X{f,g} since the mapping f ā’ 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 ā’ā’ R ā’ā’ C ā (M) ā’ā’ h(ā„¦) ā’ā’ 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 ā (M) makes it into a central extension of

the algebra of Hamiltonian vector ļ¬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 ā’ā’ h(ā„¦) ā’ā’ sp(ā„¦) ā’ā’ HdR (M, R) ā’ā’ 0,

1

where the right hand arrow is just the map described by X ā’ [X ā„¦]. Since the bracket of

two elements in sp(ā„¦) lies in h(ā„¦), 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

1

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.

Involution

I now want to make some remarks about the meaning of the Poisson bracket and its

applications.

Deļ¬nition 5: Let (M, ā„¦) be a symplectic manifold. Two functions f and g are said to

be in involution (with respect to ā„¦) if they satisfy the condition {f, g} = 0.

Note that, since {f, g} = dg(Xf ) = ā’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 otherā™s Hamiltonian

vector ļ¬eld.

Now, if one is trying to describe the integral curves of a Hamiltonian vector ļ¬eld,

Xf , the more independent functions on M that one can ļ¬nd which are constant on the

integral curves of Xf , the more accurately one can describe those integral curves. If one

were able ļ¬nd, in addition to f itself, 2nā’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 ļ¬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 ļ¬nd several functions in

involution with a given function f = f1 and, moreover, in involution with each other.

In case one can ļ¬nd nā’1 such independent functions, f2 , . . . , fn , we have the following

theorem of Liouville which says that the remaining nā’1 required functions can be found

(at least locally) by quadrature alone. In the classical language, a vector ļ¬eld Xf for which

such functions are known is said to be ācompletely integrable by quadraturesā, or, more

simply as ācompletely integrableā.

Theorem 3: Let f 1 , f 2 , . . . , f n be n functions in involution on a symplectic manifold

(M 2n , ā„¦). Suppose that the functions f i are independent in the sense that the diļ¬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

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

Moreover, the functions ai can be found by āļ¬niteā 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 ļ¬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 ļ¬elds Xf i are linearly independent on M, so by āļ¬niteā operations, we

ĀÆ ĀÆ

can construct 1-forms Ī²1 , . . . , Ī²n which satisfy the conditions

ĀÆ

Ī²i (Xf j ) = Ī“ij (Kronecker delta).

Any other set of forms Ī²i which satisfy these conditions are given by expressions:

ĀÆ

Ī²i = Ī²i + gij df j .

ńņš. 17 |