Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
Подождите немного. Документ загружается.
function
0
(a). As the fluid moves so that a 7!q, the
volume of an infinitesimal region will change, but its
mass must remain fixed. The statement of local mass
conservation is d
3
r =
0
d
3
a, where d
3
a is an initial
infinitesimal volume element that maps to d
3
q at
time t, and d
3
r = J d
3
a. (When integrating over we
will replace d
3
q by d
3
r.) Thus, we obtain
ðr; tÞ¼
0
ðaÞ
J ða; tÞ
a¼q
1
ðr;tÞ
¼
0
J
q
1
ðr; tÞ½16
where recall the Jacobian J = det(q
i
,
j
). Besides the
density, for the ideal fluid, one attaches an entropy
per unit mass, s = s
0
(a), to a fluid element, and this
quantity remains fixe d in time. In the Eulerian
description this gives rise to the entropy field
sðr; tÞ¼s
0
ðaÞj
a¼q
1
ðr;tÞ
¼ s
0
q
1
ðr; tÞ½17
One could attach other scalar, vector, etc., quan-
tities to the fluid element, but we will not pursue
this. In the usual ideal fluid closure only the above
variables are considered.
Equations [15]–[17] express the Euler–Lagrange
map. There is a natural representation of this map in
terms of the Eulerian density variables, M : = v, ,
and := s, the momentum, mass, and entropy
densities, respectively, which, as will be seen, are
variables in which the noncanonical Poisson bracket
has Lie–Poisson from.
Other Variables
Fluid mechanics is rife with variables that have been
used for its description. For example, Euler, Monge,
Clebsch, and others introduced potential representa-
tions, of varying generality, for the Euler ian velocity
field, an example being
vðr; tÞ¼r þr ½18
where the three components of v are replaced by the
functions , , and , all of which depend on (r, t).
Often reduced variables that are tailored to
specific ideal flows with less generality than those
described by , s, and v are considered. Examples
include incompressible flow with rv = 0, vortex
dynamics, including contour dynamics and point
vortex dynamics, flow governed by the
shallow-water equations, quasigeostrophy, etc. The
Hamiltonian structure in terms of these reduced
variables derives from that of the parent model in
terms of Lagrangian variables. Specific variables
may embody constraints, and understanding these
constraints, although tractable, can be a cause of
confusion. Pursuing this further is beyond the scope
here.
Hamilton’s Principle for Fluid
Lagrange, in his famous work of 1788, Me´canique
Analytique, produced in essence a variational
principle for incompressible fluid flow in terms of
Lagrangian variables. The generalization to com-
pressible flow awaited the discovery of thermody-
namics, and that is what we describe here. In
traditional mechanics nomenclature, this variational
principle is an infinite-dimensional generalization of
what is known variously as the action principle, the
principle of least action, or Hamilton’s principle,
whereby one constructs, on physical grounds, a
Lagrangian function on TQ used in the action
principle, where Q is the function space of the
q(a, t).
Construction of the Lagrangian requires identifi-
cation of the potential energy, and this requires
thermodynamics, because potential energy is stored
in terms of pressure and temperature. A basic
assumption of the fluid approximation is that of
local thermodynamic equilibrium . In the energy
representation of thermodynamics, the extensive
energy is treated as a function of the entropy and
the volume. For a fluid, it is convenient to consider
the energy per unit mass, denoted by U,tobea
function of the entropy per unit mass, s, and the
mass density, , a measure of the volume. The
intensive quantities, pressure and temperature, are
given by T = @U=@s and p =
2
@U=@. Choices for
U produce equations of state. For barotropic or
isentropic flow, U depends only on . For an ideal
monatomic gas U(, s) = c
1
exp (s), where c, ,
and are constants. The funct ion U could also
depend on additional scalar quantities, such as a
quantity known as spice that has been considered in
oceanography.
Conventional thermodynamic variables can be
viewed as Eulerian variables with a static velocity
field. Thus, we write U(, s), where and s are
spatially independent or, if the system has only locally
relaxed, these variables can be functions of r. For the
ideal fluid, each fluid element can be viewed as a self-
contained isentropic thermodynamic system that
moves with the fluid. Thus, the total fluid potential
energy functional is given by V[q] =
R
d
3
a
0
U
(s
0
,
0
=I ), which is a functional of q that depends
only upon I and hence only upon @q=@a .
The next item required for constructing Hamilton’s
principle is the kinetic energy functional, which is
given by T[q,
_
q] =
R
d
3
a
0
_
q
2
=2, where
_
q
2
:=
ij
_
q
i
_
q
j
,
with the Cartesian metric
ij
:=
ij
. This metric and its
inverse can be used to raise and lower indices.
The Lagrangian functional is L[q,
_
q]:= T V,
where L[q,
_
q] =
R
d
3
aL(q,
_
q, @q=@a) and L is the
596 Hamiltonian Fluid Dynamics
Lagrangian density, in terms of which the action
functional of Hamilton’s principle is given by
S½q¼
Z
t
1
t
0
dtL½q;
_
q
¼
Z
t
1
t
0
dt
Z
d
3
a
1
2
0
_
q
2
0
U
½19
The end conditions for Hamilton’s principle for the
fluid are the same as those of mechanics, that is,
q(a, t
0
) = q(a, t
1
) = 0. The nonpenetration condi-
tion, q
^
n = 0on@,where
^
n is a unit normal vector
is also assumed. Other boundary conditions, such as
periodic and free boundary conditions, are also
possibilities. Hamilton’s principle amounts to
S=q(a, t) = 0, which, with the end and boundary
conditions, implies the following equations of motion:
0
€
q
i
þ A
j
i
@
@a
j
2
0
I
2
@U
@
¼0 ½20
Here we have used @A
j
i
=@a
j
= 0, which can be seen
using [14]. Equation [20] amounts to Newton’s
second law for the ideal fluid, which is made clearer
by using the following useful identity:
@
@q
k
¼
1
I
A
i
k
@
@a
i
½21
Alternatively, upon using [13], [20] is sometimes
written in the form
0
€
q
j
@q
j
@a
i
þ I
@
@a
i
2
0
I
2
@U
@
¼0 ½22
The Eulerian variable force law follows from [20]
upon using [21]:
@v
@t
þ v rv
¼rp ½23
where v = v(r, t). The remaining Eulerian equati ons
of mass conservation and entropy advection follow
from the constraints that s
0
and
0
are constant on
fluid elements. Time differentiation and the trans-
formations of [16] and [17] yield
@
@t
þrðvÞ¼0 ½24
@s
@t
þ v rs ¼0 ½25
Equations [23]–[25] together with a g iven function
U(, s) and the relation p =
2
@U=@ constitute the
Eulerian description.
Variational principles similar to that described
above exist for essentially all ideal fluid models,
including incompressible flow, magnetohydrody-
namics, the two-fluid equations of plasma physics, etc.
Eulerian Action Principles
Some early researchers sought variational principles
that directly produce the ideal fluid equations in
Eulerian form. Because the Eulerian form of the
equations does not treat the fluid as a collection of
particles, the resulting action principles possess a
certain awkwardness. Below, we describe three
approaches to such action princ iples.
Clebsch action The action principle for electromag-
netism proceeds by introducing the 4-vector potential.
In a similar way, the Clebsch action principle
anticipates this idea by using a potential representation
of the velocity field, an example being that of [18].
Although compressible flow with an arbitrary
equation of state can be treated in full generality, for
simplicity and variety we will restrict to incompres-
sible flow and set rv 0. This constraint is
enforced by requiring to be dependent on and
according to [, ]:=
1
(r), where
1
is
the inverse Laplacian. The Clebsch action is then
written as follows:
S
C
½; :¼
Z
t
1
t
0
dt
Z
d
3
r
t
1
2
v
2
½26
where the subscript t denotes differentiation at fixed r,
we have set 1, and v is a shorthand for the
expression of [18] with = [, ]. The form of S
C
is
that of the phase-space action that produces Hamilton’s
equations upon independent variation of the configura-
tion space coordinate and its conjugate momentum,
which are here and , respectively. Thus, we require
(r, t
0
) = (r, t
1
) = 0, but no condition is needed for
at t
0, 1
.Wealsorequire
^
n v = 0on@.The
variations S
C
= = 0andS
C
= = 0imply
t
¼
H
¼v r
t
¼
H
¼v r ¼0
½27
an infinite-dimensional version of [1] with
H :=
R
d
3
rv
2
=2. Evidently, both and are
advected by the flow.
Because the vorticity, := rv = r r, knowl-
edge of and determines and one can invert the curl
operator to obtain v in the usual way. The intersection of
level sets of and define vortex lines, and, evidently,
these quantities, like the entropy for compressible
dynamics, are constant on fluid elements. It is not
difficult to show that the advection of and implies
the correct dynamical equation for incompressible v.
Herivel–Lin action The Herivel–Lin action incor-
porates [24] and [25] as constraints with Lagrange
Hamiltonian Fluid Dynamics 597
multipliers, ’ and . (Here is not the Clebsch
and the factor of is included for convenience.) It
was discovered early on that these constraints were
not enough to achieve complete genera lity and so a
new one, known as the Lin constraint, was added.
The Lin constraint corresponds to constancy of the
fluid particle label. One defines an Eulerian label
field by setting q(a, t) = r and solving for the label
a = q
1
(r, t) =: a(r, t). Conservation of particle identity
is thus given by a
t
þ v ra = 0, and this constraint is
associated with a Lagrange multiplier = (
1
,
2
,
3
).
The Herivel–Lin action is thus given by
S
HL
½v;;s; a; ’; ;
¼
Z
t
1
t
0
dt
Z
d
3
r
1
2
v
2
U ð; sÞþ’
t
þrðvÞ½
s
t
þv rs½ a
t
þv ra½
½28
Variation of [28] with respect to the Lagrange
multipliers just reproduces the constraints; however,
variation with respect to v, , s, and a produces
equations that imply [23]. Moreover, every flow can
be shown to be an extremal of S
HL
.
Euler–Poincare´–Hamel action Another approach is
to use directly constrained variations. The essential
idea is to only consider Eulerian variable variations
that are induced by underlying Lagrangian variable
variations q, the so-called dynamically accessible
variations. Explicitly, a basic Eulerian variation
= (
1
,
2
,
3
) is given by (r, t) = q(a, t)j
a = q
1
(r, t)
.
In terms of this quantity, the dynamically accessible
variations of the Eulerian velocity field, density,
and entropy are given, respectively, by v =
t
þ v
r rv, = r(), and s = rs . Upon
inserting them into the variation of
S
EPH
½¼
Z
t
1
t
0
dt
Z
d
3
r
1
2
v
2
Uð; sÞ
½29
and integrating by parts gives
S
EPH
¼
Z
t
1
t
0
dt
Z
d
3
r½... ¼ 0
where [ ...] is equivalent to [23]. Thus, assuming
is arbitrary, we obtain directly the equation of
motion.
There is a version of this kind of constrained
variational principle for all ideal fluid and plasma
equations. Also, it possesses a geome tric interpreta-
tion. In a more practical vein, constrained variations
can be used to derive reduced models, and dynami-
cally accessible variations can also be used for
stability calculations. Exploring these ideas is out-
side the present scope.
Fluid Hamiltonian Description
Having described variational principl es, we turn
to the associated canonical and noncanonical
Hamiltonian descriptions.
Canonical Description
Because the action of [19] is of standard form, it is
convex in
_
q and the Legendre transform follows
easily: the canonical momentum density is
i
(a, t):= L=
_
q
i
(a) =
0
_
q
i
and H[q, ] =
R
d
3
a[
_
q L] =
R
d
3
a[
2
=(2
0
) þ
0
U]. Hamilton’s equa-
tions are then
_
q
i
¼
H
i
¼fq
i
; Hg; _
i
¼
H
q
i
¼f
i
; Hg½30
an infinite-dimensional version of [1], with the
canonical Poisson bracket
fF; Gg¼
Z
F
q
G
G
q
F
d
3
a ½31
(Note, q
i
(a)=q
j
(a
0
) =
i
j
(a a
0
), a relation analo-
gous to @q
j
=@q
i
=
j
i
for finite systems.)
Reduction to Noncanonical Poisson Brackets
Reduction is a procedure for reducing the size of a
Hamiltonian system. Given constants of motion in
involution, that is, with pairwise vanishing Poisson
brackets, the dimension of a Hamiltonian system
can be reduced by 2 for each such constant of
motion. However, when constants do not commute,
the situation is more complicated and one must
invoke a theory due to Lie, Poincare´, Cartan, and
others. Associated with invariants are symmetries,
and so a complete discussion of this theory requires
examination of symmetry groups and associated
geometry. For the ideal fluid, the map from the
Lagrangian to the Eulerian descriptions is an
example of reduction, whereby the Poisson bracket
of [31] is mapped into a noncanonical Poisson
bracket. En route to describing this example, a brief
discussion of reduction of finite systems is consid-
ered first.
Reduction of Finite-Dimensional Systems
Consider a canonical system with the phase space
M,a2N-dimensional symplectic manifold. In a
coordinate patch with coordinates z = (q, p) the
system has the canonical description of [2]–[4 ].
Suppose we have a map P : M!m
, wher e m
is
some M < 2N-dimensional space described by coor-
dinates w = ( w
1
, w
2
, ..., w
M
). In coordinates, this
map is represented in terms of functions w
a
= w
a
(z),
598 Hamiltonian Fluid Dynamics
with a = 1, 2, ..., M, which, because M < 2N,is
always noninvertible. Suppose f , g : M!R obtain
their z-dependence through the funct ions w, that is,
f (z) =
f (w(z)) =
f w. Making use of the chain rule
yields
½f ; g¼
@
f
@w
a
J
ab
@
g
@w
b
½32
where the quantity
J
ab
:¼
@w
a
@z
J
c
@w
b
@z
½33
is in general a function of z. However, it is possible
that J
ab
may only depend on w. When this happens,
we have a reduction of the phase space M.
If the original dynamics of interest has the
Hamiltonian vector field generated by H(z), and if
it is possible that H(z) can be expressed solely in
terms of the w’s, that is, H(z) =
H(w), then the
system has been reduced. Clearly, this is a statement
of symmetry, since the function H(z) in reality
depends on a fewer number of variable s, the w’s.
A beautiful form of reduction occurs when the
map P has a special form w
a
= L
i
a
(q)p
i
, wher e the
quantity L is associated with a symmetry group. An
identity for what is required of L
i
a
in order for the
transformed bracket to be expressible in terms of the
w’s can be worked out, but this is explained in terms
of Lie groups. If the space m is a Lie algebra g , then
the functions
f ,
g are real-valued functions on g
that
can be extended by left or right translation to
functions
f ,
g on T
G . Thus, f restricted to T
G at
the identity, T
e
G = g
,is
f . Because T
G is a
cotangent bundle, it carries the canonical Poisson
bracket and we get a natural map P, called a
momentum map, into the dual of a Lie algebra. This
geometrical description of obt aining brackets on g
from brackets on T
G is a case of Marsden–
Weinstein reduction. In the early 1980s, these
authors and others developed the geometrical inter-
pretation of the nonc anonical Poisson brackets for
the ideal fluid.
Ideal Fluid Noncanonical Poisson Brackets
The Euler–Lagrange map of the fluid is of the form
of the map P above. It maps the canonical bracket of
[31] into a noncanonical Poisson bracket. If we use
the Eulerian variables M := v, , and := s, then
the resulting noncanonical bracket is of Lie–Poisson
form. To effect this map, one must vary [15]–[17] to
relate functional derivatives with respect to q and
to those with respect to M, , and . This amounts
to working out the chain rule for functionals. Upon
doing this, one obtains the following noncanonical
bracket:
fF; Gg¼
Z
M
i
F
M
j
@
@x
j
G
M
i
G
M
j
@
@x
j
F
M
i
þ
F
M
r
G
G
M
r
F
þ
F
M
r
G
G
M
r
F
d
3
r ½34
This bracket, together with the Hamiltonian
H[M, , ] =
R
d
3
r[M
2
=(2) þ U(, =)] generates
the ideal fluid equations. This Hamiltonian follows
from
H[M, , ]:= H[q, ] with H[q, ] =
R
d
3
a
[
2
=(2
0
) þ
0
U]. The bracket of [34] is clearly seen
to be linear in the variables M, ,and, and the form
of the cosymplectic operator and structure operators
C
k
ij
can be obtained by integration by parts. The Lie
group in this case can be seen to be an extension by
semidirect product of the diffeomorphism group.
An alternative form of the noncanonical Poisson
bracket is given in terms of the variables v, , and s.
Upon changing to these coordinates, the noncanoni-
cal Poisson bracket transforms into
fF; Gg¼
Z
F
r
G
v
G
r
F
v
þ
rv
G
v
F
v
þ
rs
F
s
G
v
G
s
F
v
d
3
r ½35
which, with the Hamiltonian H[v, , s] =
R
d
3
r
[v
2
=2 þ U(, s)], produces the Eulerian fluid equa-
tions of [23]–[25] directly as v
t
= {v, H},
t
= {, H},
and s
t
= {s, H}, respectively. Observe that in these
variables, the bracket is no longer of Lie–Poisson form.
Conclusion
In a general sense, Hamiltonian dynamics is about
coordinate changes, and it is clear from the above
that there is no shortage of coordinates for describ-
ing the ideal fluid. The most intuitive form of fluid
equations (at present) is the Eulerian form, and this
possesses a noncanonical Hamiltonian description.
Other noncanonical variables are also used for both
less and more general fluid systems than those
described above. Vortex dynamics, shallow-water
theory, and other equations of geophysical fluid
dynamics are possibilities, as well as equatio ns from
plasma physics and other disciplines. The general
story for these systems is much the same as above,
although in some descriptions constraints are
involved and they can complicate matters.
Hamiltonian Fluid Dynamics 599
There are various motivations for pursuing an
understanding of the Hamiltonian structure of
fluids, but ultimately these motivations are the
same as those for investigating the Hamiltonian
dynamics of particle and other finite degree-of-
freedom systems. Hamiltonian theory serves as an
organizing framework, one that can be used for the
derivation and approximation of systems. If one
understands something about a particular Hamilto-
nian system, then often it can be said to be true of a
general class of Hamiltonian systems. By now, many
applications have been worked out, some of which
can be accessed from the literature cited below.
See also: Adiabatic Piston; Adiabatic Piston;
Bi-Hamiltonian Methods in Soliton Theory; Bi-Hamiltonian
Methods in Soliton Theory; Classical r-Matrices, Lie
Bialgebras, and Poisson Lie Groups; Contact Manifolds;
Contact Manifolds; Hamiltonian Group Actions;
Infinite-Dimensional Hamiltonian Systems; Korteweg–de
Vries Equation and other Modulation Equations;
Korteweg–de Vries Equation and Other Modulation
Equations; Stochastic Hydrodynamics; Stochastic
Hydrodynamics.
Further Reading
Holm DD, Marsden JE, and Ratiu TS (1998) The Euler–Poincare´
equations and semidirect products with applications to
continuum theories. Advance in Mathematics 137: 1–81.
Kuznetzov EA and Zakharov VE (1997) Hamiltonian formalism
for nonlinear waves. Usp. Fiz. Nauk 167: 1137–1168.
Morrison PJ (1982) Poisson brackets for fluids and plasmas. In:
Tabor M and Treve Y (eds.) Mathematical Methods in
Hydrodynamics and Integrability in Dynamical Systems,
American Institute of Physics Conference Proceedings, No. 88.
New York: American Institute of Physics.
Morrison PJ (1998) Hamiltonian description of the ideal fluid.
Reviews in Modern Physics 70: 467–521.
Morrison PJ (2005) Hamiltonian and action principle formula-
tions of plasma physics. Physics of Plasmas 12: 058102–1–13.
Morrison PJ and Greene JM (1980) Noncanonical Hamiltonian
density formulation of hydrodynamics and ideal magneto-
hydrodynamics. Physical Review Letters 45: 790–793; (E)
(1982) 48: 569.
Salmon R (1988) Hamiltonian fluid mechanics. Annual Reviews
of Fluid Mechanics 20: 225–256.
Serrin J (1959) Mathematical principles of classical fluid
mechanics. In: Flu¨ gge S (ed.) Handbuch der Physik, vol. VIII.
Part 1, pp. 125–263. Berlin: Springer.
Shepherd TG (1990) Symmetries, conservation laws, and Hamil-
tonian structure in geophysical fluid dynamics. Advances in
Geophysics 32: 287–338.
Hamiltonian Group Actions
L C Jeffrey, University of Toronto, Toronto, ON,
Canada
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The idea of a Hamiltonian flow on a symplectic
manifold has its roots in Hamilton’s equations,
which govern the trajectory of a particle in phase
space (the space parametrizing coordinates and
momenta of a classical particle). A fundamental
idea in theoretical physics (Noether’s theorem) is
that to every symmetry in a physical system (such as
a group action), there is an associated conserved
quantity: invariance under translation corresponds
to conservation of linear momentum, invariance
under rotation corresponds to conservation of
angular momentum and so on, an d these momenta
are functions on the phase space. The mathematical
formulation of this idea is the idea of the moment
map associated to a group action on a symplectic
manifold; the group action is obtained from the
Hamiltonian flow of the moment map.
This article will describe some basic features of
moment maps associated to Hamiltonian group
actions, and some recent results about the geometry
and topology of symplectic manifolds which have such
group actions. We first define Hamiltonian group
actions and list some of their properties. Next we give
the definition of the symplectic quotient, which is a
means of dividing out the symmetry to form a new
symplectic manifold. We also explain some properties
of the quotient construction. The convexity theorem
and the moment polytope are outlined and toric
manifolds (a particular type of symplectic manifold
with a Hamiltonian torus action of maximal dimen-
sion) are defined. Finally, we list some properties of
cohomology rings of symplectic quotients.
Two standard references on this material are the
books of Cannas da Silva (2001) and McDuff and
Salamon (1995). An authoritative and comprehen-
sive reference is the monograph by Guillemin,
Ginzburg and Karshon (2002).
Hamiltonian Group Actions
Let (M, !) be a symplectic manifold. The Hamiltonian
vector field
H
generated by a function H is defined by
!
m
ð
H
; YÞ¼dH
m
ðYÞ
600 Hamiltonian Group Actions
for any Y 2 T
m
M.IfX 2 g 7!X
#
are the vector
fields on M generated by the symplectic action of a
compact Lie group G with Lie algebra g, then the
moment map : M !g
is defined by two
properties:
1. d
m
(Y)(X) = !
m
(X
#
, Y) for any Y 2 T
m
M:in
other words the function
X
: M !R defined by
X
ðmÞ¼
def
ðmÞðXÞ
is the Hamiltonian function generating the vector
field X
#
.
2. : M !g
is equivariant (where G acts on g
by
the coadjoint action).
Remark 1 In this article, we shall only consider
actions of compact connected Lie groups, although
the definition of Hamiltonian group action may be
extended to noncompact groups. In particular,
unless otherwise specified the term ‘‘torus’’ refers
to the compact torus T ffi U(1)
n
.
Remark 2 (Existence and uniqueness of moment
maps). One sees that L
X
#
! = d(
X
#
!), so that
X
#
!
is closed. The moment map
X
exists if and only if
X
# ! is also ‘‘exact.’’ The moment map need not
always exist: for example, if S
1
acts on T
2
by
e
iX
: ðe
i
1
; e
i
2
Þ7!ðe
ið
1
þXÞ
; e
i
2
Þ
we see that for the standard symplectic form
! = d
1
^ d
2
we have
X
# ! = d
2
. Since
2
is only
defined mod 2 we see that the moment map does
not exist as a map into R. Conditions guaranteeing
the existence of a moment map (other than M being
simply connected) include the hypothesis that G
is semisimple (Guillemin and Sternberg (1990,
theorem 26.1)); conditions on the existence and
uniqueness of the moment map can be formulated in
terms of Lie algebra cohomolo gy (see Guillemin and
Sternberg (1990)). The obstruction to the existenc e
of the moment map for a symplectic action of G is
an element of H
1
(g); the obstruction to uniqueness
of the moment map is an element of H
2
(g), where g
is the Lie algebra of G. See Guillemin and Sternberg
(1990, proposition 24.1).
Basic Properties of Moment Maps
Proposition 1 (Guillemin–Sternberg (1982, 1984))
Imðd
m
Þ
?
¼ LieðStabðmÞÞ
where ? denotes the annihilator under the canonical
pairing g
g !R.
Proof We have
!ðY
#
m
; ZÞ¼d
Y
ðZÞ¼hY; d
m
ðZÞi
for all Z 2 T
m
M.Thus,Y annihilates all 2 Im(d
m
)
if and only if Y 2 Lie(Stab(m)).
Corollary 1 Zero is a regular value of if and only
if Stab(m) is finite for all m 2
1
(0). In this
situation,
1
(0) is a manifold and the stabilizer of
the action at any point in
1
(0) is finite.
Example 1 Let T be a torus acting on M and let
F M
T
be a component of the fixed-point set. Then
for any f 2 F, we have d
f
= 0, so (F) is a point.
Proposition 2
(i) If H G are two groups acting in a Hamiltonian
fashiononasymplecticmanifoldM, then
H
=
G
where : g
!h
is the projection map. In
other words, if X 2 h, then
H
(m)(X) =
G
(m)(X)
for any m 2 M. One example that frequently
arises is the case when H = T is a maximal torus of
a compact Lie group G.
(ii) More generally if f : H !G is a Lie group
homomorphism, and the two groups G and H
act in a Hamiltonian fashion on a symplectic
manifold M, in such a way that the action is
compatible with the homomorphism f, then
H
= f
G
where f
: g !h
is induced from
the homomorphism f. ( The case (i) is the special
case where f is the inclusion map.)
(iii) If two symplectic manifolds M
1
and M
2
are acted
on in a Hamiltonian fashion by a group G with
moment maps
1
and
2
, then the moment map
for the diagonal action of G on M
1
M
2
with the
product symplectic structure is
1
þ
2
.
Example 2 The standard symplectic form on S
2
is
! = dcos ^ d = dz ^ d (where is the polar
angle, is the azimuthal angle, and z is the height
function). The associated moment map for the action
of U(1) on S
2
by rotation about the z axis is (z, ) = z.
Example 3 If R
2
= C has the symplectic structure
! = dx ^ dy, the moment map for the standard
action of U(1) on R
2
with multiplicity m 2 Z,in
other words the action
u 2 Uð1Þ : z 2 C 7!u
m
z
is (x, y) = m (x
2
þ y
2
)=2.
Example 4 Suppose a torus T acts on C preserving
the standard symplectic structure, and suppose
the action factors through a homomorphism
B: T !U(1) which can be written as
Bðexp
T
XÞ¼exp
Uð1Þ
ððXÞÞ
Hamiltonian Group Actions 601
in terms of a linear map 2 t
that maps the integer
lattice of t into Z (in other words, a weight) and the
exponential maps
exp
T
: t ! T
and
exp
Uð1Þ
: R ! Uð1Þ
(the latter being normalized as exp
U(1)
(t) = e
2it
).
Then, by Proposition 2(ii) and Example 3 we
see that the moment map for the action of T on
C is
ðzÞ¼
1
2
jzj
2
It follows that if T acts on C
n
via a collection of
weights
1
, ...,
n
2 t
, then the moment map is
ðz
1
; ...; z
n
Þ¼
1
2
X
n
j¼1
jz
j
j
2
j
and the image of the moment map is the cone in t
spanned by {
1
, ...,
n
}.
The Symplectic Quotient
Since the moment map is equivariant, we may
form the symplectic quotient (or Marsden–Weinstein
reduction)
M
red
¼ M
0
¼
1
ð0Þ=G
The symplectic structure on M descends to give a
symplectic structure on M
0
. Corollary 1 implies that
if 0 is a regular value of , then M
0
is an orbifold.
Remark 3 Another way to formulate Corollary 1 is
that if G acts freely, 0 is a regular value of so
1
(0) is a manifold with a free G action, and hence
1
(0)=G is also a manifold. If the G action is only
locally free, then
1
(0) is still a manifold, but the
quotient
1
(0)=G is only an orbifold.
Remark 4 The definition of orbifold is due to
Satake (1957); an alternate form ulation is given in
the paper by Henriques and Metzler (2004) and
references cited there.
If T is a torus, then the equivariance condition on
the moment map reduces to invariance, so we may
form the reduced space M
t
=
1
(t) =T for any
regular value t 2 t
of the moment map ; the
space M
t
is a symplectic orbifold for any regular
value t of .
Example 5 Let U(1) act diagonally on C
n
equipped
with the standard symplectic structure
! ¼
i
2
X
n
j¼1
dz
j
^ d
z
j
¼
X
n
j¼1
dx
j
^ dy
j
½1
where z
j
= x
j
þ iy
j
. The moment map for this action is
ðz
1
; ...; z
n
Þ¼
1
2
X
n
j¼1
jz
j
j
2
so the symplectic quotient
1
(1=2)=U(1) is com-
plex projective space
S
2n1
=Uð1ÞffiCP
n1
More generally we may consider the reduced
space M
=
1
(O
)=G when O
is the orbit in g
through 2 g
(coadjoint orbit). All such orbits may
be parametrized by 2 t
þ
, where t
þ
is a chosen
positive Weyl chamber in t
.
Example 6 Let U(n) act on C
n
in the standard way,
where C
n
is equippe d with the standard symplectic
structure [1]. The moment map for this action is
ðz
1
; ...; z
n
Þ
jk
¼
i
2
z
j
z
k
½2
which is the (j, k) element of a matrix in the Lie
algebra of U(n). The standard symplectic form on
C
n
descends under reduction to the standard
symplectic form on CP
n1
(which corresponds to
the Fubini–Study metric).
Example 7 (Coadjoint orbits). Let 2 g
.We
define a symplectic structure !
on the coadjoint
orbit O
(in terms of the vecto r fields X
#
, Y
#
generated by the action of X, Y 2 g)by
!
(X
#
, Y
#
) = ([X, Y]) at the point 2O
(and
everywhere else on the orbit by equivariance). The
moment map for the action of G on O
with respect
to this symplectic structure is the inclusion of O
in g
. (The symplectic structure on the orbit was
found by Kirillov and Kostant; see, for instance,
Berline et al. (1992, section 7.5).
Example 8 (The shifting trick). Define a symplectic
structure on M O
by
¼ !
M
!
Then for the moment map with respect to the
induced action of G on M O
we have
M
ffiðM O
Þ
0
602 Hamiltonian Group Actions
Corollary 2 Combining Example 6 with Proposition
2(ii) we see that for any linear action of a group G on
CP
n1
(i.e., an action factoring through a representa-
tion G !U(n), or in other words an action descending
from a linear action on C
n
) the moment map factors as
¼
^
where
ˆ
: CP
n1
!u(n)
is given in [3] below, and
: u(n)
!g
is the projection map.
In particular, one often requires for a projective
manifold M (i.e., a compact complex manifold with
an embedding into CP
n1
) that the action of G
extends to a linear action on CP
n1
. Thus, moment
maps for such linear actions are given by [3]
composed with and with the embedding of M
into CP
n1
(see also Cotangent Bundle Reduction,
Poisson Reduction, Symmetry and Symplectic
Reduction).
Reduction in Stages
Suppose a compact Lie group G acts in a Hamilto-
nian fashion on a symplectic manifold M, and H is a
normal subgroup of G. (For example, this hypoth-
esis is satisfied if both H and G are tori.) Suppose
also that 0 is a regular value for
H
and
G
. Then
the symplectic quotient
1
H
(0)=H is acted on
naturally by the quotient group G=H, and this
action is Hamiltonian; furthermore, the symplectic
quotient of
1
H
(0)=H by G=H is naturally iso-
morphic to
1
G
(0)=G. (Th is result is known as
‘‘reduction in stages.’’)
Let M be a symplectic manifold equipped with the
Hamiltonian action of a torus T. Let H T be a Lie
subgroup of T (so H is a torus whose dimension is
smaller than the dimension of T). Let
T
: M !
Lie(T)
and
H
: M !Lie(H)
be the moment maps:
recall that
H
=
H
T
, where
H
: Lie(T)
!
Lie(H)
is the standard projection.
For any 2 Lie(H)
we may form the reduced
space M
=
1
H
()=H. This is equipped with a
Hamiltonian action of T=H.
Example 9 Let U(n) act on C
n
in the standard way.
This action descends to an action on CP
n1
, which
is the symplectic quotient of C
n
under the action of
the diagonal U(1) subgroup of U(n). Hence, the
moment map
ˆ
for the action of U(n)onCP
n1
is
given by the formula
^
ð½z
1
; ...; z
n
Þ
jk
¼
i
2
z
j
z
k
P
n
‘¼1
jz
‘
j
2
½3
which comes from the moment map [2] for the
action of U(n)onC
n
.
The Normal Form Theorem
There is a neighborhood of
1
(0) on which the
symplectic form is given in a standard way related to
the symplectic form !
0
on M
red
(see, e.g., Guillemin
and Sternberg (1990, sections 39–41)).
Proposition 3 (Normal form theorem). Assume 0 is
a regular value of (so that
1
(0) is a smooth
manifold and G acts on
1
(0) with finite stabilizers).
Then there is a neighborhood U ffi
1
(0) {z 2 g
,
jzjh}
1
(0) g
of
1
(0) on which the sym-
plectic form is given as follows. Let P
def
=
1
(0)
q
!
M
red
be the orbifold principal G-bundle
given by the projection map q :
1
(0) !
1
(0)=G,
and let 2
1
(P) g be a connection for it. Let !
0
denote the induced symplectic form on M
red
, in other
words q
!
0
= i
0
!. Then if we define a 1-form on
U P g
by
p, z
= z()(for p 2 Pandz2 g
), the
symplectic form on U is given by
! ¼ q
!
0
þ d ½4
Further, the moment map on U is given by
(p, z) = z.
Corollary 3 Let t be a regular value for the
moment map for the Hamiltonian action of a torus
T on a symplectic manifold M. Then in a neighbor-
hood of t, all symplectic quotients M
t
are diffeo-
morphic to M
t
0
by a diffeomorphism under which
!
t
= !
t
0
þ (t t
0
,d) where 2
1
(
1
(t
0
)) t is a
connection for the action of T on
1
(t
0
).
Corollary 4 Suppose G acts in a Hamiltonian
fashion on a symplectic manifold M, and suppose
0 is a regular value for the moment map . Then the
reduced space M
=
1
(O
)=G at the orbit O
fibers over M
0
=
1
(0)=G with fiber the orbit O
;
furthermore, if : M
!M
0
is the projection map,
then the symplectic form !
on
1
(O
)=G is give n
as !
=
!
0
þ
, where !
0
is the symplectic form
on M
0
and
restricts to the standard Kirillov–
Kostant symplectic form on the fiber.
Convexity Theorems
Theorem 1 (Atiyah (1982); Guillemin–Sternberg
(1982 and 1984)). Suppose M is a connected
compact symplectic manifold equipped with a Hamil-
tonian action of a torus T. Then the image (M) is a
convex polytope, the convex hull of {(F)}, where F are
the components of the fixed-point set of T in M.
Example 10 Consider the orbits O
t
of SU(2) in
su(2) ffi R
3
through t 2 R
þ
. The image of the
moment map for the action of the maximal torus
T ffi U(1) is the interval [t, t].
Hamiltonian Group Actions 603
Example 11 When O
t
is the coadjoint orbit
(through t 2 t
) for a compact Lie group G with
maximal torus T, the image
T
(O
t
) of the moment
map
T
for the action of the maximal torus T is the
convex hull Conv{wt : w 2 W}, where W is the Weyl
group.
The convexity theorem above can be generalized
to actions of nonabelian groups. If M is a connected
compact symplectic manifold equipped with a
Hamiltonian actio n of a compact Lie group G with
maximal torus T and positive Weyl chamber t
þ
,
then the intersection of the image (M)ofthe
moment map with the positive Weyl chamber t
þ
(in
other words, a fundamental domain for the action
of the Weyl group on t) is a convex polytope.
This result is due to Kirwan (1984b) and for Ka¨ hler
manifolds to Guillemin and Sternberg (1982 and
1984).
The proofs of Atiyah and Guillemin–Sternberg are
based on Morse theory applied to the moment map.
A key ingredient in the proofs is to establish that the
fibers of the moment map are connected.
The Moment Polytope
Given a compact symplectic manifold M equipped
with the Hamiltonian action of a torus T, we see
that there is an associated polytope P, the ‘‘moment
polytope.’’ The fibers of the moment map are
preserved by the action of T, so the value of
parametrizes a family {M
t
} of symplectic quotients.
By Theorem 1 the moment polytope is the convex
hull of the images of the fixed-point set under the
moment map.
By Proposition 1, we see that the moment
polytope is decomposed according to the stabilizers
of points in the preimage, and the critical values of
the moment map are the images
T
(W
j
) of the
fixed-point sets W
j
of one-parameter subgroups S
j
of T. These critical values form hyperplanes
(‘‘walls’’) which subdivide the moment polytope:
the complement of the walls is a collection of open
regions consisting of regular values of the moment
map.
Example 12 The group SU(3) has maximal torus
T ffi U(1)
2
. We identify g
with g via the bi-invariant
inner product (i.e., the Killing form) on g, and thus
identify t
with t. For 2 t, the Weyl group images
of are the six vertices of a hexagon: the ‘‘walls’’ in
the moment polytope for the action of T on the
coadjoint orbit O
arising from the action of G on
g
through 2 t
are the edges of the hexagon
(exterior walls) and the three lines connecting
opposite vertices (interior walls).
Toric Manifolds
Definition 1 A toric manifold is a compact
symplectic manifold M of dimension 2n equipped
with the effective Hamiltonian action of a torus T of
dimension n.
Example 13 Complex projective space CP
n
with
the obvious Hamilton ian action of U(1)
n
U(1)
nþ1
is a toric manifold.
Example 14 A special case of Example 13 is the
2-sphere S
2
ffi CP
1
(with the action of U(1) given by
rotation around one axis). The 2-sphere is a toric
manifold.
Elementary Properties of Toric Manifolds
If M is a toric manifold, the fiber of the moment map
for the action of T is an orbit of the action. Hence,
the symplectic quotient M
t
at any value t 2 t
is a
point (if it is nonempty).
The regular values of are the inte rior points of
the moment polytope P. All points in the preimage
1
(@P) are fixe d points of some one-parameter
subgroup of T. Points in the interior of a face P
j
of
dimension j are fixed by a subtorus of T of
dimension n j. Hence, each fiber of over a
point in P
j
is a quotient torus of dimension j.In
particular, the vertices of the polytope are the
images of the components of the fixed-point set of
the whole torus T, and the inverse image of a vertex
is contained in the fixed-point set of T.
The pus h-forward function
(!
n
=n!) under the
moment map is just the characteristic function of the
moment polytope.
Delzant’s Theorem
In fact, toric manifolds are characterized by their
moment polytopes. A theorem of Delzant (1988)
says that any polytope P satisfying appropriate
hypotheses (a simple polytope) is the moment
polytope for some toric manifold; furthermore, if
two toric manifolds acted on effectively by a torus T
have the same moment polytope, then they are
T-equivariantly symplectomorphic. The first state-
ment is proved by constructing a toric manifold
which has the polytope P as its moment polytope; if
P has d faces of codimension 1, one constructs the
toric manifold M as a sympl ectic quotient of a
vector space V ffi C
d
by the linear action of a torus
T
0
ffi U(1)
dn
. The torus T ffi U(1)
n
acting on M is
then obtained by reduction in stages, as the quotient
of U(1)
d
by T
0
.
The construction of a toric manifold whose
moment polytope is a given simple polytope is
604 Hamiltonian Group Actions
given in Guillemin (1994, chapter 1). The second
statement (namely that toric manifolds are classified
by their moment polytopes) is proved in Delzant
(1988).
Example 15 The moment polytope for the action
of U(1)
n
on CP
n
is the n-simplex. This actio n
descends from the action of U(1)
nþ1
on C
nþ1
, using
reduction in stages: recall from Example 5 that we
constructed CP
n
as the symplectic quotient of C
nþ1
by the standard action of U(1).
Cohomology Rings of Symplectic
Quotients
For material on the equivariant cohomology of
symplectic manifolds equipped with Hamiltonian
group actions and the relation to the fixed-point set,
we refer to Equivalent Cohomology and the Cartan
Model. As in that reference we shall describe
the equivariant cohomology of a Hamiltonian
G-manifold using the Cartan model.
Two fundamental results of Kirwan give comple-
mentary descriptions of the equivariant cohomol ogy
of a symplectic manifold.
Kirwan Injectivity
Kirwan’s first theorem is the injectivity theorem:
Theorem 2 (Injectivity theorem). If T is a compact
torus and M is a Hamiltonian T-space, then the
direct sum of restriction maps to all components of
the fixed-point set
F
: H
T
ðMÞ!H
ðFÞSðt
Þ
is injective.
The proof appears in Kirwan (1984a); this
material is treated in Equivalent Cohomology and
the Cartan Model (theorem 6.6).
Kirwan Surjectivity
Let G be a complex torus, and let 0 be a regular
value of the moment map . Suppose M is a
compact symplectic manifold equipped with a
Hamiltonian action of a compact Lie group G.
There is a natural map : H
G
(M) !H
(M
red
)
defined by
: H
G
ðMÞ7!H
G
ð
1
ð0ÞÞ ffi H
ðM
red
Þ
(where the first map is the restriction map and the
second is the identification of H
G
(Z) with H
(Z=G)
when G acts locall y freely on Z and the cohomology
is taken with rational coefficients). The map is
obviously a ring homomorphism. Kirwan’s second
theorem treats the image of .
Theorem 3 (Surjectivity theorem). Under the
above hypotheses, the map is surjective.
The proof of this theorem (Kirwan (1984a, 5.4 and
8.10); see also Kirwan (1992, section 6)) uses the
Morse theory of the ‘‘Yang–Mills function’’ jj
2
:
M !R to define an equivariant stratification of M
by strata S
which flow under the gradient flow of
jj
2
to a critical set C
of jj
2
. One shows that the
function jj
2
is equivariantly perfect (i.e., that the
Thom–Gysin (long) exact sequence in equivariant
cohomology decomposes into short exact sequences,
so that one may build up the cohomology as
H
G
ðMÞffiH
G
ð
1
ð0ÞÞ
M
6¼0
H
G
ðS
Þ
Here, the stratification by S
has a partial order >;
thus, one may define an open dense set U
= M
[
>
S
which includes the open dense stratum S
b
of
points that flow into
1
(0) (note S
b
retracts onto
1
(b)). The equivariant Thom–Gysin sequence is
!H
n2dðÞ
G
ðS
Þ!
i
H
n
G
ðU
Þ!H
n
G
ðU
S
Þ!
To show that the Thom–Gysin sequence splits into
short exact sequences, it suffices to know that the
maps (i
)
are injective. Since i
(i
)
is multiplication
by the equivariant Euler clas s e
of the normal
bundle to S
, injectivity follows because this
equivariant Euler class is not a zero divisor (see
Kirwan (1984a, 5.4) for the proof).
Because is a surj ective ring homomorphism, it
follows that
H
G
ðM
red
ÞffiH
T
ðMÞ=KerðÞ
The above theorem is also valid when G is the
complexification of a compact semisimple Lie
group. In this case, one must reduce at 0 (because
of the condition that the moment map is equivar-
iant, since b = 0 is the only value which is invariant
under the coadjoint action). The case of reducing at
coadjoint orbits can be treated using the proof for
the case of reducing at 0 via the shifting trick
(Example 8).
Several recent articles (Jeffrey and Kirwan (1995,
1997), Tolman and Weitsman 2003) compute
Ker(). Some articles compute Ker() in specific
examples, notably the action of S
1
on products of
two-dimensional spheres of general radii.
The Residue Formula
One approach to identifying Ker() is the ‘‘residue
formula,’’ Jeffrey and Kirwan (1995), theorem 8.1:
Hamiltonian Group Actions 605