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characterize the dynamical behavior of these models

(Robertson 1933) and find exact and approximate

solutions to these equations as well as phase planes

representing the relation of the different models to

each other; for example, Ehlers and Rindler (1989)

give the phase planes for models with noninteracting

matter and radiation and an arbitrary cosmological

constant. Universes with maxima or minima in S(t)

can only occur if k = þ1; when =0, the universe

recollapses in the future iff k = þ1. Static solutions

are possible only if k = þ1 and (assuming [4])

 > 0. The simplest expanding solutions are the

Einstein–de Sitter universes with k = 0 =.

Equation [8] is a special case of [6],with

corresponding implications: if the combined matter

present satisfies [4], with   0, then there must have

been an initial singularity, or at least the universe

must have emerged from a quantum gravity domain.

The temperature would have been arbitrarily high in

the past, so there was a hot big bang era in the early

universe where matter and radiation were in equili-

brium with each other at very high temperatures that

rapidly fell as the universe expanded. Many physical

processes took place then, in particular nucleosynth-

esis of light elements took place at 10

K. Decou-

pling of matter and radiation took place at a

temperature of 4000 K, followed by formation of

stars and galaxies (see Peacock (1999) for a discus-

sion of these physical processes). The black-body

radiation emitted by the surface of last scattering at

4000 K is observed by us today as cosmic black-body

radiation (CBR) at a temperature of 2.75 K.

One can determine observational relations for

these models such as the magnitude–redshift relation

for ‘‘standard candles’’ at recent times from the EFEs

(Sandage 1961). The aim of observations is to

determine the Hubble constant H

, dimensionless

deceleration parameter q

= (3=H

)(

€

S=S)

,and

normalized density parameters 

= 

=3H

for

each component of matter present. The spatial

curvature and the cosmological constant then follow

from [6] and [9]; also the present scale factor S

determined if k 6¼ 0. The universe is of positive

spatial curvature (k = þ1) iff 

 

þ 



> 1,

where 





, 



==3H

. Current observa-

tions indicate 

’ 0.3, 



’ 0.7, 

’ 1.02 

0.02. Because the nucleosynthesis results limit the

baryon density to a very low value (

’ 0.02),

which is about the same as the density of luminous

matter, this indicates the dominant presence of both

nonbaryonic dark matter and a repulsive force

corresponding to either a cosmological constant or

varying scalar field (dark energy).

Crucial causal limitations occur because of the

existence of particle horizons (Rindler 1956), the

nature of which is most clear when represented in

conformal diagrams (Hawking and Ellis 1973, Tipler

et al. 1980). These result from the fact that light

can only proceed a finite distance in the finite time

since the origin of the universe, and imply that for

a standard radiation-dominated hot-big-bang early

universe, regions of larger than 1



angular size on

the surface of last scattering, which emits the CBR,

are causally disconnected: hence, no causal process

since the start of the universe can account for the

extreme isotropy of the CBR (T=T ’ 10

5

over

the whole sky, once a dipole anisotropy T=T ’

3

due to our local velocity relative to the

cosmological rest frame is allowed for). This is the

‘‘horizon problem,’’ one of the driving forces

behind the theory of ‘‘inflation’’ (Guth 1981): the

idea that, in the very early universe, a slow-rolling

scalar field led to a brief exponential expansion

through at least 50 e-folds (during which time the

spacetime was approximately de Sitter), thus

smoothing the universe and solving the horizon

problem (Guth 1981, Peacock 1999). This is

possible because a scalar field can violate the energy

condition [3] and so allows acceleration:

€

S > 0.

Consequently, there are now many studies of the

dynamics of FLRW solutions driven by scalar fields

and the subsequent decay of these scalar fields into

radiation. One interesting point is that one can

obtain exact solutions of this kind for arbitrarily

chosen evolutions S(t), provided they satisfy a

restriction on the magnitude of

, by running the

field equations backwards to determine the needed

potential V()(Ellis and Madsen 1991). The

inflationary paradigm is dominant in present-day

theoretical cosmology, but suffers from the problem

that it is not in fact a well-defined theory, for there

is no single accepted proposal for the physical

nature of the effective scalar field underlying the

supposed exponential expansion; rather there are

numerous competing proposals. As the inflaton has

not yet been identified, this theory is not yet

soundly linked to well-established physics.

Approximate FL Solutions

The real universe is, of course, not exactly FL, and

studies of structure formation depend on studies of

solutions that are approximately FL models – they

are realistic (‘‘lumpy’’) universe models. These

enable detailed studies of observable properties

such as CBR anisotropies and gravitational lensing

induced by matter inhomogeneities, and of the

development of those inhomogeneities from quan-

tum fluctuations in the very early universe that then

get expanded to very large scales by inflation.

Cosmology: Mathematical Aspects 655

The key problem here is that apart from the standard

coordinate freedom allowed in general relativity, there

is a serious gauge issue: the background FL model is not

uniquely determined by the realistic universe model;

however, the magnitudes of many perturbed quantities

depend on how it is fitted into the lumpy model. For

example, the density perturbation  is determined

pointwise by the equation

ðx

Þðx

Þðx

where

(x

) is the background density. But by

altering the correspondence between the background

and realistic models (specifically, by the choice of

surfaces

(x

) = const. in the realistic model) one can

assign that quantity any value, including zero (if one

chooses

(x

) = (x

)). This is the ‘‘gauge problem.’’

One can handle it by using standard variables and

keeping close track of the gauge freedom at all

times. However, one then ends up with higher-order

equations than necessary because some of the

perturbation modes present are pure gauge modes

with no physical significance. Alternatively, one can

fix the gauge by some unique specification of how

the background model is fitted into the realistic

model, but there is no agreement on a unique way to

do this, and different choices give different answers.

The preferable resolution is to use gauge-invariant

variables, either coordinate based (Bardeen 1980)or

covariant, based on the (1þ3) covariant decomposi-

tion of spacetime quantities mentioned above (Ellis

and Bruni 1989), in either case resulting in pertur-

bation equations without gauge freedom and of

order corresponding to the physical degrees of

freedom. The key point in the latter approach is to

choose covariant variables that vanish in the back-

ground spacetime; they are then automatically gauge

invariant. Realistic structure formation studies carry

out this process for a mixture of matter components

with different average velocities, and extend to a

kinetic theory description of the background radia-

tion (see Ellis and van Elst (1999) and references

therein). The outcome is a prediction of the CBR

anisotropy power spectrum, determined by the

inhomogeneities in the gravitational field and the

motions of the matter components at decoupling

(Sachs and Wolfe 1967). This spectrum can then be

compared with observations and used in determin-

ing the values of the cosmological parameters

mentioned above (see Peacock 1999).

One crucial issue is why it is reasonable to use a

perturbed FL model for the observable region of the

universe. The key argument is that this is plausible

because of the high isotropy of all observations

around us when averaged on a sufficiently large

spatial scale, and particularly the very low anisotropy

of the CBR. The Ehlers–Geren–Sachs (EGS) theorem

(Ehlers et al. 1968) provides a sound basis for this

argument: it shows that if freely propagating CBR

(obeying the Liouville equation) is exactly isotropic in

an expanding universe domain U,then the universe is

exactly FL in that domain (i.e., it has exactly the RW

spatially homogenous and isotropic geometry there),

the point being that any inhomogeneities in the

matter distribution between us and the surface of last

scattering will produce anisotropies in the CBR

temperature we measure. But that result does not

apply to the real universe, because the CBR is not

exactly isotropic. The ‘‘almost EGS’’ theorem

(Stoeger et al. 1995) shows that this result is stable:

almost isotropic CBR in the domain U implies that

the universe is almost-FL in that domain. The

application to the real universe comes by making a

weak Copernican assumption: ‘‘we assume we are

not special, so all observers in U (taken to be the

visible part of the universe) will also see almost

isotropic CBR, just as we do.’’ The result then

follows. A further argument for homogeneity of the

universe comes from postulating ‘‘uniform thermal

histories’’ (Bonnor and Ellis 1986), but that argument

is yet to be completed and applied in a practical way.

Anisotropic and Inhomogeneous Models

The FL universes are geometrically extremely special.

We wish further to understand the full range of

possible universe models, their dynamical behaviors,

and which of them might, at some epoch, realistically

represent the real universe. This enables us to see how

the approximate FL models fit into this wider set of

possibilities, and under what circumstances they are

attractors in this set of cosmologies.

Exact solutions are characterized by their space-

time symmetries. Symmetries are characterized by

the dimension s of the surfaces of homogeneity and

the dimension q of the isotropy group at a general

point, together giving the dimension r = s þ t of the

group of isometries G

(at special points, such as a

center of symmetry, s can decrease and q increase

but always so that r stays unchanged). In the case of

a cosmological model, because the 4-velocity u

invariant under isotropies, the only possible dimen-

sions for the isotropy group are q = 3, 1, 0; whereas

the dimension t of the surfaces of homogeneity can

take any value from 4 to 0. This gives the basis for a

classification of cosmological spacetimes (Ellis 1967,

Ellis and van Elst 1999).

When q = 3, we have isotropic solutions – there

are no preferred spatial directions – and it is then

a theorem that they must be spatially homoge-

neous FL universes (Ehlers 1961). When q = 1, we

656 Cosmology: Mathematical Aspects

have locally rotationally symmetric (LRS) solu-

tions, with precisely one preferred spacelike direc-

tion at a generic point (Ellis 1967). When q = 0, the

solutions are anisotropic in that there can be no

continuous group of rotations leaving the solution

invariant; however, there can be discrete isotropies

in some special cases.

When t = 4,we have spacetime homogeneous solu-

tions, with all physical quantities constant; they cannot

expand (by [5] and [3]). Nevertheless, two cases are of

interest. For q = 1(r = 5) we find the Go¨ del universe,

rotating everywhere with constant vorticity, which

illustrates important causal anomalies (Go¨del 1949,

Hawking and Ellis 1973). For q = 3(r = 6), we find

the Einstein ‘‘static universe’’ (Einstein 1917), the

unique nonexpanding FL model with k = 1and > 0.

It is of interest because it could possibly represent the

asymptotic initial state of nonsingular inflationary

universe models (Ellis et al. 2003). The higher-

symmetry models (de Sitter and anti-de Sitter

universes with higher-dimensional isotropy groups)

are not included here because they do not obey the

energy condition [3] – they are empty universes,

which can be interesting asymptotic states but are

not by themselves good cosmological models.

When t = 3, we have spatially homogeneous

evolving universe models. For q = 0(r = 3), there

are a large family of Bianchi universes, spatially

homogeneous but anisotropic, characterized into

nine types according to the structure constants of

the Lie algebra of the three-dimensional symmetry

group G

. These can be ‘‘orthogonal’’: the fluid flow

is orthogonal to the surfaces of homogeneity, or

‘‘tilted’’; the latter case can have fluid rotation or

acceleration, but the former cannot. They exhibit a

large variety of behaviors, including power-law,

oscillatory, and nonscalar singularities (Tipler et al.

1980). A vexed question is whether truly chaotic

behavior occurs in Bianchi IX models. The behavior

of large families of these models has been character-

ized in dynamical systems terms (Wainwright and

Ellis 1996), showing the intriguing way that higher-

symmetry solutions provide a ‘‘skeleton’’ that guides

the behavior of lower-symmetry solutions in the

space of spacetimes. Many Bianchi models can be

shown to isotropize at late times, particularly if

viscosity is present; thus, they are asymptotic to the

FL universes in the far future. In some cases, Bianchi

models exhibit intermediate isotropization: they are

much like FL models for a large part of their life, but

are very different from it both at very early and very

late stages of their evolution. These could be good

models of the real universe. An important theorem

by Wald (1983) shows that a cosmological constant

will tend to isotropize Bianchi solutions at late

times. This is an indication that inflation can

succeed in making anisotropic early states resemble

FL models at later times. Observational properties

like element abundances and CBR anisotropy

patterns can be worked out in these models (some

of them develop a characteristic isolated ‘‘hot spot’’

in the CBR sky). For q = 1(r = 4), we have spatially

homogeneous LRS models, either Kantowski Sachs

or Bianchi universes, and again observations can be

worked out in detail and phase planes developed

showing their dynamical behavior, often isotropiz-

ing at late times. There are orthogonal and tilted

cases, the latter possibly involving nonscalar singu-

larities. For q = 3(r = 6), we have the isotropic FL

models, discussed above. Both the LRS and isotropic

cases could be good models of the real universe.

When t = 2, we have inhomogeneous evolving

models. This is a very large family, but the LRS

(q = 1, r = 2) cases have been examined in detail; in

the case of pressure-free matter, these are the

Tolman–Bondi inhomogeneous models (Bondi

1947) that can be integrated exactly, and have

been used for many interesting astrophysical and

cosmological studies. Krasin

ski (1997) gives a very

complete catalog of these and lower-symmetry

inhomogeneous models and their uses in cosmology.

A considerable challenge is the dynamical systems

analysis for generic inhomogeneous models, needed

to properly understand the early evolution of generic

universe models (Uggla et al. 2003), and hence to

determine what is generic behavior.

The Origin of the Universe

The issue underlying all this is what led to the initial

conditions for the universe, for example, providing

the starting conditions for inflation. There are many

approaches to studying the quantum gravity phase

of cosmology, including the Wheeler–de Witt equa-

tion, the path-integral approach, string cosmology,

pre-big bang theory, brane cosmology, the ekpyrotic

universe, the cyclic universe, and loop quantum

gravity approaches. These lie beyond the purview of

the present article, except to say that they are all

based on unproven extrapolations of known physics.

The physically possible paths will become clearer as

the nature of quantum gravity is elucidated.

It is pertinent to note that there exist nonsingular

realistic cosmological solutions, possible in the light

of the violations of the energy condition enabled by

the supposed scalar fields that underlie inflationary

universe theory. These nonsingular solutions can even

avoid the quantum gravity era (Ellis et al. 2003).

However, they have very fine-tuned initial conditions,

which is nowadays considered as a disadvantage; but

Cosmology: Mathematical Aspects 657

there is no proof that whatever processes led to the

existence of the universe preferred generic rather than

fine-tuned conditions; this is a philosophical rather

than physical assumption. It may well be that, as

regards the start of the universe, the options are that

either an initial singularity occurred, or the initial

conditions were very finely tuned and allowed an

infinitely existing universe. Investigation of whether

this conjecture is in fact valid, and if so which is the

best option, are intriguing open topics.

See also: Einstein Equations: Exact Solutions;

Einstein–Cartan Theory; General Relativity: Experimental

Tests; General Relativity: Overview; Gravitational

Lensing; Lie Groups: General Theory; Newtonian Limit of

General Relativity; Quantum Cosmology; Shock Wave

Refinement of the Friedman–Robertson–Walker Metric;

Spacetime Topology, Causal Structure and Singularities;

String Theory: Phenomenology.

Further Reading

Bardeen JM (1980) Physical Review D 22: 1882.

Bondi H (1947) Monthly Notices of the Royal Astronomical

Society 107: 410.

Bonnor WB and Ellis GFR (1986) Monthly Notices of the Royal

Astronomical Society 218: 605.

Ehlers J (1961) Abh Mainz Akad Wiss u Lit (translated in Gen

Rel Grav 25: 1225, 1993).

Ehlers J, Geren P, and Sachs RK (1968) Journal of Mathematical

Physics 9: 1344.

Ehlers J and Rindler W (1989) Monthly Notices of the Royal

Astronomical Society 238: 503.

Einstein A (1917) Sitz Ber Preuss Akad Wiss (translated in The

Principle of Relativity, 1993). Dover.

Ellis GFR (1967) Journal of Mathematical Physics 8: 1171.

Ellis GFR (1971) In: Sachs RK (ed.) General Relativity and

Cosmology, Proc. Int. School of Physics ‘‘Enrico Fermi,’’

Course XLVII, p. 104. Academic Press.

Ellis GFR and Bruni M (1989) Physical Review D 40: 1804.

Ellis GFR and van Elst H (1999) In: Lachieze-Ray M (ed.) Theoretical

and Observational Cosmology, vol. 541 [gr-qc/9812046], Nato

Series C: Mathematical and Physical Sciences: Kluwer.

Ellis GFR and Madsen M (1991) Classical and Quantum Gravity

8: 667.

Ellis GFR, Murugan J, and Tsagas CG (2003) gr-qc/0307112.

Ellis GFR, Nel SD, Stoeger W, Maartens R, and Whitman AP

(1985) Physics Reports 124(5 and 6): 315.

Go¨ del K (1949) Reviews of Modern Physics 21: 447.

Guth A (1981) Physical Review D 23: 347.

Hawking SW and Ellis GFR (1973) The Large Scale Structure of

Space Time. Cambridge: Cambridge University Press.

Krasin

ski A (1997) Inhomogeneous Cosmological Models.

Cambridge: Cambridge University Press.

Kristian J and Sachs RK (1966) The Astrophysical Journal 143: 379.

Lachieze-Rey M and Luminet JP (1995) Physics Reports

254: 135–214.

Robertson HP (1933) Reviews of Modern Physics 5: 62.

Peacock JA (1999) Cosmological Physics. Cambridge: Cambridge

University Press.

Rindler W (1956) Monthly Notices of the Royal Astronomical

Society 116: 662.

Sachs RK and Wolfe A (1967) Astrophysical Journal 147: 73.

Sandage A (1961) Astrophysical Journal 133: 355.

Stoeger W, Maartens R, and Ellis GFR (1995) Astrophysical

Journal 443: 1.

Tipler FJ, Clarke CJS, and Ellis GFR (1980) In: Held A (ed.)

General Relativity and Gravitation: One Hundred Years after

the Birth of Albert Einstein, vol. 2, p. 97. Plenum.

Uggla C, van Elst H, Wainwright J, and Ellis GFR (2003)

Physical

Review D gr-qc/0304002 (to appear).

Wainwright J and Ellis GFR (eds.) (1996) The Dynamical Systems

Approach to Cosmology. Cambridge: Cambridge University

Press.

Wald RM (1983) Physical Review D 28: 2118.

Cotangent Bundle Reduction

J-P Ortega, Universite

de Franche-Comte

Besanc¸on, France

T S Ratiu, Ecole Polytechnique Fe

rale

de Lausanne, Lausanne, Switzerland

Introduction

The general symplectic reduction theory (see

Symmetry and Symplectic Reduction) becomes

much richer and has many applications if the

symplectic manifold is the cotangent bundle



Q, 

= d

) of a manifold Q. The canonical

1-form 

on T



Q is given by 

(

)(V



) =







)), for any q 2Q, 



Q, and

tangent vector V





Q), where 

: T



Q !Q

is the cotangent bundle projection and T









Q) !T

Q is its tangent map (or derivative)

at q. In natural cotangent bundle coordinates (q

, p

we have 

= p

and 

= dq

^ dp

Let  : G  Q !Q be a left smooth action of the Lie

group G on the manifold and Q. Denote by

g  q =(g, q) the action of g 2G on the point q 2Q

and by 

: Q !Q the diffeomorphism of Q induced

by g. The lifted left action G  T



Q !T



Q,givenby

g  

= T



gq



1

(

)forg 2G and 



preserves 

, and admits the equivariant momentum

map J : T



Q !g



whose expression is hJ(

), i=



((

(q)), where  2g ,theLiealgebraofG, h, i: g





g !R is the duality pairing between the dual g



and g ,

and 

(q) = d(expt, q)=dtj

t = 0

is the value of the

658 Cotangent Bundle Reduction

infinitesimal generator vector field 

of the G-action

at q 2Q (se e Hamiltonian Group Actions and

Symmetries and Conservation Laws). Throughout

this article, it is assumed that the G-action on Q,

and hence on T



Q,isfreeandproper.Recallalso

that ((T





,(

)



) denotes the reduced manifold

at  2g



(see Symmetry and Symplectic Reduction),

where (T





:= J

1

()=G



is the orbit space of the



-action on the momentum level manifold J

1

()

and G



:= {g 2G j Ad



 = } is the isotropy sub-

group of the coadjoint representation of G on g



The left-coadjoint representation of g 2G on  2g



is denoted by Ad



1

.

Cotangent bundle reduction at zero is already quite

interesting and has many applications. Let  : Q !Q=G

be the G-principal bundle projection defined by the

proper free action of G on Q, usually referred to as the

shape space bundle. Zero is a regular value of J and the

map ’

:((T



,(

)

) !(T



(Q=G), 

Q=G

)given

by ’

([

])(T

(v

)) := 

), where 

1

(0),

[

] 2(T



,andv

Q, is a well-defined sym-

plectic diffeomorphism.

This theorem generalizes in two nontrivial ways

when one reduces at a nonzero value of J:an

embedding and a fibration theorem.

Embedding Version of Cotangent

Bundle Reduction

Let  2g



, Q



:= Q=G



, 



: Q !Q



the projection

onto the G



-orbit space, g



:= { 2g jad





 = 0} the

Lie algebra of the coadjoint isotropy subgroup G



where ad



 := [, ] for any ,  2g ,ad





: g



the

dual map, 

:= j







the restriction of  to g



and ((T





,(

)



) the reduced space at .The

induced G



-action on T



Q admits the equivariant

momentum map J



: T



Q !g





given by J



(

) =

J(



. Assume there is a G



-invariant 1-form 



on Q with values in ( J



)

1

(

). Then there is a unique

closed 2-form 



on Q



such that 









= d



.Define

the magnetic term B



:= 









,where







is the cotangent bundle projection,

which is a closed 2-form on T





.Thenthemap

’



:((T





,(

)



) !(T





, 



 B



)givenby

’



([

])(T





)):= (





(q))(v

), for 

1

(),

[

]2(T





,andv

Q, is a symplectic embed-

ding onto a submanifold of T





covering the base



. The embedding ’



is a diffeomorphism onto





if and only if g =g



.Ifthe1-form



takes

values in the smaller set J

1

() then the image of ’



the the vector sub-bundle [T



(VQ)]



of T





, where

VQ TQ is the vertical vector sub-bundle consisting

of vectors tangent to the G-orbits, that is, its fiber at

q2Q equals V

Q={

(q) j2g }, and



denotes the

annihilator relative to the natural duality pairing

between TQ



and T





.Notethatifg is abelian or

=0, the embedding ’



is always onto and thus the

reduced space is again, topologically, a cotangent

bundle.

It should be noted that there is a choice in this

theorem, namely the 1-form 



. Whereas the

reduced symplectic space ((T





,(

)



) is intrin-

sic, the symplectic structure on the space T





depends on 



. The theorem above states that no

matter how 



is chosen, there is a symplectic

diffeomorphism, which also depends on 



, of the

reduced space onto a submanifold of T





Connections

The 1-form 



is usually obtained from a left

connection on the principal bundle 



: Q !Q = G



 : Q !Q=G. A left connection 1-form A 2

(Q; g )

on the left principal G-bundle  : Q !Q=G is a Lie

algebra-valued 1-form A : TQ !g,whereg denotes

the Lie algebra of G, satisfying the conditions A(

) = 

for all  2g and A(T



(v)) = Ad

(A(v)) for all g 2G

and v 2T

Q,whereAd

denotes the adjoint action of

G on g. The horizontal vector sub-bundle HQ of the

connection A is defined as the kernel of A,thatis,its

fiber at q 2Q is the subspace H

:= ker A(q). The map

7!ver

):= [A(q)(v

)]

(q) is called the vertical

projection, while the map v

7!hor

):= v



ver

) is called the horizontal projection. Since for

any vector v

Q we have v

= ver

) þ hor

it follows that TQ = HQ  VQ and the maps

hor

: T

Q ! H

Q and ver

: T

Q !V

Q are projec-

tions onto the horizontal and vertical subspaces at every

q 2Q.

Connections can be equivalently defined by the

choice of a sub-bundle HQ TQ complementary to

the vertical sub-bundle VQ satisfying the following

G-invariance property: H

gq

Q = T



Q) for

every g 2G and q 2Q. The sub-bundle HQ is called,

as before, the horizontal sub-bundle and a connection

1-form A is defined by setting A(q)(

(q) þ u

) = ,

for any  2g and u

The curvature of the connection A is the Lie

algebra-valued 2-form on Q defined by B(u

, v

) =

dA(hor

), hor

)). When one replaces vectors in

the exterior derivative with their horizontal projec-

tions, then the result is called the exterior covariant

derivative and the preceding formula for B is often

written as B = DA. Curvature measures the lack of

integrability of the horizontal distribution, namely

B(u, v) = A([hor(u), hor(v)]) for any two vector

fields u and v on Q. The Cartan structure equations

state that B(u, v) = dA(u, v)  [A(u), A(v)], where

the bracket on the right hand side is the Lie

bracket in g .

Cotangent Bundle Reduction 659

Since the connection A is a Lie algebra-valued

1-form, for each  2g



the formula 



(q):=

A(q)



(), where A(q)



: g



Q is the dual of the

linear map A(q):T

Q !g , defines a usual 1-form on

Q. This 1-form 



takes values in J

1

() and is

equivariant in the following sense: 





= 





for

any g 2G.

Magnetic Terms and Curvature

There are two methods to construct the 1-form 



from a connection. The first is to start with a

connection 1-form A



2

(Q; g



) on the principal



-bundle 



: Q !Q=G



. Then the 1-form 



hj



, A



i2

(Q)isG



-invariant and has values in

( J



)

1

(j



). The magnetic term B



is the pullback to



(Q=G



) of the j



-component d



of the

curvature of A



thought of as a 2-form on the

base Q=G



The second method is to start with a connection

A 2

(Q, g ) on the principal bundle  : Q !Q=G,

to define 



:= h, Ai2

(Q), and to observe that

this 1-form is G



-invariant and has values in J

1

().

The magnetic term B



is in this case the pullback to



(Q=G



)ofthe-component d



of the curvature

of A thought of as a 2-form on the base Q=G



The Mechanical Connection

If (Q, hh, ii) is a Riemannian manifold and G acts by

isometries, there is a natural connection on the

bundle  : Q !Q=G, namely, define the horizontal

space at a point to be the metric orthogonal to the

vertical space. This connection is called the mechan-

ical connection and its horizontal bundle consists of

all vectors v

2TQ such that J(hhv

, ii) = 0.

To determine the Lie algebra-valued 1-form A of

this connection, the notion of locked inertia tensor

needs to be introduced. This is the linear map

I(q):g !g



depending smoothly on q 2Q defined by

the identity hI(q), i= hh

(q), 

(q)ii for any

,  2g . Since the G-action is free, each I(q)is

invertible. The connection 1-form whose horizontal

space was defined above is given by A(q)(v

) =

I(q)

1

( J(hhv

, ii)).

Denote by K : T



Q !R the kinetic energy of the

metric hh, ii on the cotangent bundle, that is,

K(hhv

, ii):= (1=2)kv

. The 1-form 



= A(  )



 is

characterized for the mechanical connection A by the

condition K(



(q)) = inf {K(

) j 

2 J

1

() \ T



Q}.

The Amended Potential

A simple mechanical system is a Hamiltonian system

on a cotangent bundle T



Q whose Hamiltonian

function is the the sum of the kinetic energy of a

Riemannian metric on Q and a potential function

V : Q !R. If there is a Lie group G acting on Q by

isometries and leaving the potential invariant, then

we have a simple mechanical system with symmetry.

The amended or effective potential V



: Q !R at

 2g



is defined by V



:= H  



, where 



is the

1-form associated to the mechanical connection. Its

expression in terms of the locked moment of inertia

tensor is given by V



(q):= V(q) þ (1=2) , I(q)

1





The amended potential naturally induces a smooth

function



(Q=G



The fundamental result about simple mechanical

systems with symmetry is the following. The push-

forward by the embedding ’



:((T





,(

)



) !





, 



 B



) of the reduced Hamiltonian



((T





) of a simple mechanical system

H = KþV  



Q) is the restriction to the

vector sub-bundle ’



((T





)  T



(Q=G



), which

is also a symplectic submanifold of (T



(Q=G





Q=G



 B



), of the simple mechanical system on



(Q=G



) whose kinetic energy is given by the

quotient Riemannian metric on Q=G



and whose

potential is



. However, Hamilton’s equations on



(Q=G



) for this simple mechanical system are

computed relative to the magnetic symplectic form



Q=G



B



There is a wealth of applications starting from

this classical theorem to mechanical systems, span-

ning such diverse areas as topological characteriza-

tion of the level sets of the energy–momentum map

to methods of proving nonlinear stability of relative

equilibria (block-diagonalization of the stability

form in the application of the energy–momentum

method).

Fibration Version of Cotangent Bundle Reduction

There is a second theorem that realizes the reduced

space of a cotangent bundle as a locally trivial

bundle over shape space Q=G. This version is

particularly well suited in the study of quantization

problems and in control theory. The result is the

following. Assume that G acts freely and properly

on Q. Then the reduced symplectic manifold (T





is a fiber bundle over T



(Q=G) with fiber the

coadjoint orbit O



. How this is related to the

Poisson structure of the quotient (T



Q)=G will be

discussed later.

The Kaluza–Klein Construction

The extra term in the symplectic form of the reduced

space is called a magnetic term because it has this

interpretation in electromagnetism. To understand

why B



is called a magnetic term, consider the

problem of a particle of mass m and charge e

moving in R

under the influence of a given

660 Cotangent Bundle Reduction

magnetic field B = B

i þ B

j þ B

k, divB = 0. The

Lorentz force law (written in the International

System) gives the equations of motion

¼ ev  B ½1

where e is the charge and v = (

z) =

q is the

velocity of the particle. What is the Hamiltonian

description of these equations?

There are two possible answers to this question.

To formulate them, associate to the divergence free

vector field B the closed 2-form B = B

dy ^ dz

dx ^ dz þ B

dx ^ dy. Also, write B = curl A for

some other vector field A = (A

, A

)onR

called the magnetic potential.

Answer 1 Take on T



the symplectic form



= dx ^ dp

þ dy ^ dp

þ dz ^ dp

 eB, where

, p

) = p := mv is the momentum of the

particle, and h = mkvk

=2 = m(

)=2isthe

Hamiltonian, the kinetic energy of the particle. A

direct verification shows that dh =

,  ), where

þ eðB

y  B

zÞ

þ eðB

z  B

xÞ

þ eðB

x  B

xÞ

½2

which gives the equations of motion [1].

Answer 2 Take on T



the canonical symplec-

tic form =dx ^ dp

þ dy ^ dp

þ dz ^ dp

and the

Hamiltonian h

= kp  eAk

=2m. A direct verifica-

tion shows that dh

=(X

,  ), where X

has the

same expression [2].

Next we show how the magnetic term in the

symplectic form 

is obtained by reduction from

the Kaluza–Klein system. Let Q = R

 S

with

the circle G = S

acting on Q, only on the second

factor. Identify the Lie algebra g of S

with R. Since

the infinitesimal generator of this action defined

by  2g = R has the expression 

(q, ) = (q, ; 0, ),

if TS

is trivialized as S

 R, a momentum

map J : T



Q = R

S

R

R !g



= R is given by

J(q,; p,p) =(p,p) (0,)=p, that is, J(q,; p,p)=p.

In this case, the coadjoint action is trivial, so for any

 2g



= R, we have G



= S

, g



= R, and 

= . The

1-form 



= (A

dx þ A

dy þ A

dz þ d) 2

(Q),

where d denotes the length 1-form on S

, is clearly



= S

-invariant, has values in J

1

() = {(q, ; p, ) j

q, p 2R

,  2S

}, and its exterior differential equals

d



= B. Thus, the closed 2-form 



on the base



= Q=G



= Q=S

= R

equals  B and hence

the magnetic term, that is, the closed 2-form



= 









on T





= T



,isalsoB since





: Q = R

 S

!Q= G



= R

is the projection.

Therefore, the reduced space (T





symplectically diffeomorphic to (T



, dx ^ dp

dy ^ dp

þ dz ^ dp

 B), which coincides with the

phase space in Answer 1 if we put  = e.Thisalso

gives the physical interpretation of the momentum

map J : T



Q = R

 S

 R

 R !g



= R, J(q, ;

p, p) = p and hence of the variable conjugate to

the circle variable  : p represents the charge.

Moreover, the magnetic term in the symplectic

form is, up to a charge factor, the magnetic field.

The kinetic energy Hamiltonian

hðq;; p; pÞ :¼

kpk

of the Kaluza–Klein metric, that is, the Riemannian

metric obtained by keeping the standard metrics on

each factor and declaring R

and S

orthogonal,

induces the reduced Hamiltonian



ðqÞ¼

kpk



which, up to the constant 

=2, equals the kinetic

energy Hamiltonian in Answer 1. Note that this

reduced system is not the geodesic flow of the

Euclidean metric because of the presence of the

magnetic term in the symplectic form. However,

the equations of motion of a charged particle in a

magnetic field are obtained by reducing the geodesic

flow of the Kaluza–Klein metric.

A similar construction is carried out in Yang–

Mills theory where A is a connection on a principal

bundle and B is its curvature. Magnetic terms also

appear in classical mechanics. For example, in

rotating systems the Coriolis force (up to a dimen-

sional factor) plays the role of the magnetic term.

Reconstruction of Dynamics

for Cotangent Bundles

A general reconstruction method of the dynamics

from the reduced dynamics was given in (see

Symmetry and Symplectic Reduction). For cotangent

bundles, using the mechanical connection, this

method simplifies considerably.

Start with the following general situation. Let G act

freely on the configuration manifold Q;leth : T



Q !

R be a G-invariant Hamiltonian,  2g



, 

2 J

1

(),

and c



(t) the integral curve of the reduced system with

initial condition [

] 2(T





given by the reduced

Hamiltonian function h



:(T





!R. In terms of a

connection A 2

( J

1

(); g



) on the left G



-principal

bundle J

1

() ! (T





the reconstruction procedure

proceeds in four steps:



Step 1: Horizontally lift the curve c



(t) 2(T





to a curve d(t) 2 J

1

() with d(0) = 



Step 2: Set (t) = A(d(t))(X

(d (t))) 2g



Cotangent Bundle Reduction 661



Step 3: With (t) 2g



determined in step 2, solve

the nonautonomous differential equation

g(t) =

g(t)

(t) with initial condition g(0) = e, where L

denotes left translation on G; this is the step that

involves ‘‘quadratures’’ and is the main obstacle

to finding explicit formulas.



Step 4: The curve c(t) = g(t)  d(t), with d(t) found

in step 1 and g(t) found in step 3 is the integral

curve of X

with initial condition c(0) = 

This method depends on the choice of the conne-

ction A 2

( J

1

(); g



). Here are several particular

cases when this procedure simplifies.

(a) One-dimensional coadjoint isotropy group.If



= S

or G



= R, identify g



with R via the map

a 2R $a 2g



, where  2g



,  6¼ 0, is a generator of



. Then a connection 1-form on the S

(or R)

principal bundle J

1

() !(T





is the 1-form A on

1

() given by A = (1=h, i)



, where 



is the

pullback of the canonical 1-form  2



Q)to

the submanifold J

1

(). The curvature of this

connection is the 2-form on (T





given by

curv(A) = (1=h, i)!



, where !



is the reduced

symplectic form on (T





. In this case, the curve

(t) 2g



in step 2 is given by (t) =[h](d(t)), where

 2X(T



Q) is the Liouville vector field character-

ized by the property of being the unique vector field

on T



Q that satisfies the relation d(, ) = .In

canonical coordinates (q

, p

)onT



Q, =p

(b) Induced connection. Any connection A2



(Q; g



) on the left principal bundle Q !Q=G



induces a connection A 2

( J

1

(); g



)byA(

)



):= A(q)(T







)), where q 2Q, 

2 T



Q), and 

: T



Q !Q is the cotangent

bundle projection. In this case, the curve (t) 2g



step 2 is given by (t) = A(q(t))(Fh(d(t)), where

q(t):= 

(d(t)) is the base integral curve and the

vector bundle morphism Fh : T



Q !TQ is the fiber

derivative of h given by

Fhð

Þð

Þ:¼



t¼0

hð

þ t

for any 

, 



Q. Two particular instances of

this situation are noteworthy.

(b1) Assume that the Hamiltonian h is that of a

simple mechanical system with symmetry.

Choosing A to be the mechanical connection

mech

, the curve (t) 2g



in step 2 is given by

(t) = A

mech

(q(t )) (hhd(t), ii).

(b2) If Q = G is a Lie group, dim G



= 1, and  is a

generator of g



, then the connection A 2

(G)

can be chosen to equal A(g):= (1=h, i)



1

(), where  is a generator of g



and R

is right translation on G.

mechanical systems with symmetry. The case of

simple mechanical systems with symmetry deserves

special attention since several steps in the recon-

struction method can be simplified. For simple

mechanical systems, the knowledge of the base

integral curve q(t) suffices to determine the entire

integral curve on T



Q. Indeed, if h = K þ V  

the Hamiltonian, the Legendre transformation

Fh : T



Q !TQ determines the Lagrangian system

on TQ given by ‘(u

) = (1=2)ku

 V(u

), for

Q. Lagrange’s equations are second-order

and thus the evolution of the velocities is given by

the time derivative

q(t) of the base integral curve.

Since Fh = (F‘)

1

, the solution of the Hamiltonian

system is given by F‘(

q(t)). Using the explicit

expression of the mechanical connection and the

notation given in the general procedure, the method

of reconstruction simplifies to the following steps.

To find the integral curve c(t) of the simple mecha-

nical system with G-symmetry h = K þ V  



Q with initial condition c(0) = 



Q, know-

ing the integral curve c



(t) of the reduced Hamil-

tonian system on (T





given by the reduced

Hamiltonian function h



:(T





!R with initial

condition c



(0) = [

] one proceeds in the follow-

ing manner. Recall the symplectic embedding

’



:((T





,(

)



) !(T



(Q=G



), 

Q=G



 B



). The

curve ’



(t)) 2T



(Q=G



) is an integral curve of

the Hamiltonian system on (T



(Q=G



), 

Q=G



 B



)

given by the function that is the sum of the kinetic

energy of the quotient Riemannian metric and the

quotient amended potential



. Let q



(t):=



Q=G



(t)) be the base integral curve of this system,

where 

Q=G



: T



(Q=G



) !Q=G



is the cotangent

bundle projection.



Step 1: Relative to the mechanical connection

mech

2

(Q; g



), horizontally lift q



(t) 2Q=G



to a curve q

(t) 2Q passing through q

(0) = q.



Step 2:Determine(t) 2g



from the algebraic system

hh(t)

(t)), 

(t))ii= h, i for all  2g



where hh, ii is the G-invariant kinetic energy

Riemannian metric on Q.Thisimpliesthat

(0)

and (0)

(q) are the horizontal and vertical compo-

nents of the vector 

]

Q which is associated by

the metric hh, ii to the initial condition 



Step 3: Solve

g(t) = T

g(t)

(t)inG



with initial

condition g(0) = e.



Step 4: The curve q(t):= g(t)  q

(t), with q

(t)

and g(t) determined in steps 2 and 4, respectively,

is the base integral curve of the simple mechanical

system with symmetry defined by the function h

satisfying q(0) = 0. The curve (Fh)

1

(

q(t)) 2T



is the integral curve of this system with initial

662 Cotangent Bundle Reduction

condition c(0) = 

. In addition, q

(t) = g(t) 

(

(t) þ (t)

(t))) is the horizontal plus vertical

decomposition relative to the connection induced

on J

1

() !(T





by the mechanical connection

mech

2

(Q; g



There are several important situations when

step 3, the main obstruction to an explicit solution

of the reconstruction problem, can be carried out.

We shall review some of them below.

(c1) ThecaseG



= S

.IfG



is abelian, the equation in

step3hasthesolutiong(t) = exp

(s)ds.If,in

addition, G



= S

,then(s) can be explicitly

determined by step 2. Indeed, if  2g



is a

generator of g



,writing(s) = a(s) for some

smooth real-valued function a defined on some

open interval around the origin, the algebraic

equation in step 2 implies that hha(s)(t)

(t)),



(t))ii=h,i,whichgivesa(s)=h,i =

k

(s))k

. Therefore, the base integral curve of

the solution of the simple mechanical system with

symmetry on T



Q passing through q is

qðtÞ¼exp h; i

k

ðq

ðsÞÞk



 q

ðtÞ

and

qðtÞ¼exp h; i

k

ðq

ðsÞÞk





ðtÞþ

h; i

k

ðq

ðsÞÞk



ðq

ðtÞÞ

(c2) The case of compact Lie groups. An obvious

situation when the differential equation in step 3

can be solved is if (t) =  for all t, where  is a

given element of g



. Then the solution is

g(t) = exp(t ). However, step 2 puts certain

restrictions under this hypothesis, because it

requires that hh(t)

(t)), 

(t))ii= h, i

for any  2g



. This is satisfied if there is a

bilinear nondegenerate form ( , )ong satisfy-

ing (, ) = hh 

(q), 

(q)ii for all q 2Q and

,  2g . This implies that ( , ) is positive

definite and invariant under the adjoint action

of G on g, so semisimple Lie algebras of

noncompact type are excluded. If G is com-

pact, which ensures the existence of a positive

adjoint invariant inner product on g ,and

Q = G, this condition implies that the kinetic

energy metric is invariant under the adjoint

action. There are examples in which such

conditions are natural, such as in Kaluza–

Klein theories. Thus, if G is a compact Lie

group and (  ,  ) is a positive-definite metric

invariant under the adjoint action of G on g

satisfying (, ) = hh

(q), 

(q)ii for all q 2Q

and ,  2g , then the element (t) in step 2 can

be chosen to be constant and is determined by

the identity (, ) = j



on g



. The solution of

the equation on step 3 is then g(t) = exp(t).

(c3) The case when

(t) is proportional to (t). Try

to find a real-valued function f(t) such that

g(t) = exp(f (t)(t)) is a solution of the equation

g(t) = T

g(t)

(t) with f (0) = 0. This gives, for

small t, the equation

f (t)(t) þ f (t)

(t) =  (t),

that is, it is necessary that (t) and

(t)be

proportional. So, if

(t) = (t)(t) for some

known smooth function (t), then this gives

f (t) =

exp(

(r)dr)ds.

(c4) ThecaseofG



solvable.Writeg(t) = exp(f

(t)

)

exp(f

(t)

) exp(f

(t)

), for some basis

{

, 

, ..., 

}ofg



and some smooth real-valued

functions f

, i = 1, 2, ..., n, defined around zero. It

is known that if G



is solvable, the equation in

step 3 can be solved by quadratures for the f

Reconstruction Phases for Simple Mechanical

Systems with S

Symmetry

Consider a simple mechanical system with symmetry

G on the Riemannian manifold (Q, hh, ii) with

G-invariant potential V 2C

(Q). If  2g



, let V



be the amended potential and



(Q=G



) the

induced function on the base. Let c : [0, T] !T



Q be

an integral curve of the system with Hamiltonian

h = K þ V  

and suppose that its projection



: [0, T] !(T





to the reduced space is a closed

integral curve of the reduced system with Hamil-

tonian h



. The reconstruction phase associated to

the loop c



(t) is the group element g 2G



, satisfying

the identity c(T) = g  c(0). We shall present two

explicit formulas of the reconstruction phase for the

case when G



= S

. Let  2g



= R be a generator of

the coadjoint isotropy algebra and write c(T) =

exp(’)  c (0); in this case, ’ is identified with the

reconstruction phase and, as we shall see in concrete

mechanical examples, it truly represents an angle.

If G



= S

, the G



-principal bundle 



: J

1

() !





:= J

1

()=G



admits two natural connec-

tions: A = (1=)



2

( J

1

()), where 



is the

pullback of the canonical 1-form on the cotangent

bundle to the momentum level submanifold J

1

(),

and 



mech

2

( J

1

()). There is no reason to

choose one connection over the other and thus there

are two natural formulas for the reconstruction

phase in this case. Let c



(t) be a periodic orbit of

period T of the reduced system and denote also by



the value of the Hamiltonian function on it.

Cotangent Bundle Reduction 663

Assume that D is a two-dimensional surface in





whose boundary is the loop c



(t). Since the

manifolds (T





and T



(Q=S

) are diffeomorphic

(but not symplectomorphic), it makes sense to

consider the base integral curve q



(t) obtained by

projecting c



(t) to the base Q=S

, which is a closed

curve of period T. Denote by



i:¼



ðq



ðtÞÞdt

the average of



over the loop q



(t). Let q

(t) 2Q

be the A

mech

-horizontal lift of q



(t)toQ and let  be

the A

mech

-holonomy of the loop q



(t) measured from

q(0), the base point of c (0); its expression is given by

exp  = exp(

B), where B is the curvature of the

mechanical connection. Denote by !



the reduced

symplectic form on (T





. With these notations the

phase ’ is given by

’ ¼





2ðh



h



iÞT



¼ þ 

k

ðq

ðsÞÞk

½3

The first terms in both formulas are the so-called

geometric phases because they carry only geometric

information given by the connection, whereas the

second terms are called the dynamic phases since

they encapsulate information directly linked to the

Hamiltonian. The expression of the total phase as a

sum of a geometric and a dynamic phase is not

intrinsic and is connection dependent. It can even

happen that one of these summands vanishes. We

shall consider now two concrete examples: the free

rigid body and the heavy top.

Reconstruction Phases for the Free Rigid Body

The motion of the free rigid body is a geodesic with

respect to a left-invariant Riemannian metric on

SO(3) given by the moment of inertia of the body.

The phase space of the free rigid body motion is



SO(3) and a momentum map J : T



SO(3) !R

the lift of left translation to the cotangent bundle is

given by right translation to the identity element.

We have identified here so(3) with R

by the

Lie algebra isomorphism x 2(R

,  ) 7!

x 2(so(3),

[ , ]), where

x(y) = x  y, and so(3)



with R

the inner product on R

. The reduced manifold

1

()=G



is identified with the sphere S

kk

in R

radius kk with the symplectic form !



= dS=kk,

where dS is the standard area form on S

kk

and G



ﬃ

is the group of rotations around the axis . These

concentric spheres are the coadjoint orbits of the Lie–

Poisson space so(3)



and represent the level sets of the

Casimir functions that are all smooth functions of

kk

,where 2R

denotes the body angular

momentum.

The Hamiltonian of the rigid body on the Lie–

Poisson space T



SO(3)=SO(3) ﬃ R

is given by

hðÞ:¼





where I

, I

> 0 are the principal moments of

inertia of the body. Let I := diag(I

, I

) denote the

moment of inertia tensor diagonalized in a principal-

axis body frame. The Lie–Poisson bracket on R

given by {f , g}() =   (rf () rg()) and the

equation of motions are

= , where  2R

the body angular velocity given in terms of  by



:==I

, for i = 1, 2, 3, that is, =I

1

. The

trajectories of the these equations are found by

intersecting a family of homothetic energy ellipsoids

with the angular momentum concentric spheres. If

> I

, one immediately sees that all orbits are

periodic with the exception of four centers (the two

possible rotations about the long and the short

moment of inertia axis of the body), two saddles

(the two rotations about the middle moment of

inertia axis of the body), and four heteroclinic orbits

connecting the two saddles.

Suppose that (t) is a periodic orbit on the sphere

kk

with period T. After time T, by how much has

the rigid body rotated in space? The answer to this

question follows directly from [3]. Taking  = =kk

and the potential v  0 we get

’ ¼ þ



kk

2kIðsÞk

ððsÞIðsÞÞðtr IÞ

ððsÞIðsÞÞ

þkk

ððsÞIðsÞÞ

where D is one of the two spherical caps on S

kk

whose boundary is the periodic orbit (t), h



is the

value of the total energy on the solution (t), and 

is the oriented solid angle, that is,

:¼

kk



; jj¼

areaD

kk

Reconstruction Phases for the Heavy Top

The heavy top is a simple mechanical systems with

symmetry S

on T



SO(3) whose Hamiltonian function

is given by h(

):= (1=2)k

]

þ Mg‘k  h, where

h 2SO(3), 



SO(3), k is the unit vector of the

spatial Oz axis (pointing in the direction opposite to

664 Cotangent Bundle Reduction