On the number of weakly Noetherian constants of motion of nonholonomic systems

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of weakly Noetherian constants of motion of nonholonomic systems Francesco Fassò

∗

, Andrea Giacobbe

†

and NicolaSansonetto

‡

(December23,2009)

Abstract

Wedevelopamethodtogiveanestimateonthenumberoffunctionallyindependentconstants

ofmotionofanonholonomicsystemwithsymmetrywhichhavethesocalled`weakly

Noethe-rian'property [22]. We show thatthis numberis boundedfromabove by thecorank ofthe

involutiveclosureofacertaindistributionontheconstraintmanifold. Theeectivenessofthe

methodisillustratedonseveralexamples.

Keywords: Nonholonomicsystems,Constantsofmotion,Firstintegrals,Noethertheorem.

MSC:37J15,37J05,70F25.

1 Introduction

Howmany(functionallyindependent)constantsofmotiondoesanonholonomicsystemwith

sym-metryhave? In these terms,this questionis certainlytoovague to be answered. Forholonomic

systemswithsymmetry,anaturalanswerisprovidedbytheconservationoftheenergymomentum

map,whichimpliesalowerboundonthenumberofconstantsofmotion. Inthenonholonomiccase,

instead,thesituation is notascleareven inthe simplestcase ofsystemswithlinearconstraints,

naturalLagrangians(=kineticminuspotentialenergies)andlifted actions,whichisthecasethat

weconsiderin thispaper. Therearetwooppositereasonsforthis:

•

On theonehand, onlycertain componentsof themomentum map areconstantsof motion, andtheir numbermaybedicult toassess. Infact, acomponentof themomentum map is

conserved if and only ifits innitesimal generatoris a section of acertain distribution, the

socalledreactionannihilatordistribution

R

◦

,see[23].

•

Ontheotherhand,theclassofconstantsofmotionofnonholonomicsystemswithsymmetry may be larger than just the components of the momentum map. In particular, this class

mayinclude functionslinearin themomentaknownasgaugemomenta. Theseare,roughly

speaking, constants of motion generated by certain vector elds which are tangent to the

∗

Università diPadova, Dipartimento diMatematica Pura e Applicata, Via Trieste 63, 35121 Padova, Italy.

(E-mail: fasso@math.unipd.it)

†

Università diPadova, Dipartimento diMatematica Pura e Applicata, Via Trieste 63, 35121 Padova, Italy.

(E-mail: giacobbe@math.unipd.it)

‡

Università di Verona, Dipartimento diInformatica,Cà Vignal2, Strada LeGrazie15, 37134 Verona, Italy.

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grouporbits but are notinnitesimal generatorsof the groupaction, see [5, 33, 22]. There

aredierentwaysofcharacterizingthevectoreldswhichproduceconservedgaugemomenta

[28,16,39,22],but heretoo,nogeneralwayofassessingtheirnumberisknown.

Inthissituation,itisverydiculttoanswerinfullgeneralitythequestionraisedabove. Therefore,

inthispaperwetrytoprovidearst,verypartialanswertoit,byproducingabound fromabove

on thenumberof all functionally independent constantsof motionwith acertain propertythe

socalledweak Noetherianpropertyidentied in[22].

Inordertoappreciatetheoriginofthisclassofconstantsofmotionrecallthat,inHamiltonian

mechanics,oneofthefundamentalpropertiesofthelinksymmetriesconservationlawsisthefact

thatthemomentummapofaHamiltonianactiondependsonlyonthegroupaction. Therefore,the

conservation of themomentum map provides conserved quantities forall systemswith invariant

Hamiltonian. This propertyis sometimes called theNoetherian property (or condition) of the

momentummap [35].

Innonholonomicmechanics,however,whichcomponentsofthemomentummapareconserved

quantities depends on the Hamiltonian (and on the constraint manifold), and the Noetherian

propertyisbroken. Infact,itisnotevencompletelyclearwhata`Noetherianconstantofmotion'

shouldbeinthenonholonomiccontext(seetheRemarkattheendofSection3.A).Amongfunctions

linearinthemomenta,someconservedcomponentsofthemomentummap(particularlythosewith

`horizontal'innitesimalgenerators,iftheyexist)dohavetheNoetherianproperty,butingeneral

gauge momenta do not. However, certain gauge momentaparticularly those with horizontal

generatorshaveaweakerproperty: they are conserved quantities of all nonholonomicsystems

with xed constraint manifold and xed kinetic energy, but anyinvariantpotentialenergy [22].

ThisistheweakNoetherian property.

In[22],thispropertywasdenedonlyforconstantsofmotionwhicharelinearinthemomenta,

whichisthecaseofthemomentummapsofliftedactionsandismostcommonlymetinexamples.

Inthisarticle,weconsiderinsteadallweaklyNoetherianconstantsofmotion(eventhenonlinear

ones,if theyexist) withthe aimof establishingatechniquewhich cangivesome informationon

the numberof them which are functionally independent. The reason whyour method doesnot

applytothelinearweaklyNoetherianconstantsofmotionalonewillbecomeclearlater.

Specically,wewillshowthat,underacertainsmoothnesshypothesis,thenumberof

function-allyindependent smoothlocal weaklyNoetherian constantsof motionof anonholonomicsystem

whoseHamiltonianisinvariantunderaliftedactionisgivenbythecorankoftheinvolutiveclosure

∆

∞

ofacertaindistribution

∆

ontheconstraintmanifold. Wewillgiveacompletedescriptionof

∆

in thecaseoffreeand properactions,and wewill recallastandardtechniqueto computethe corankof

∆

∞

if

∆

isrealanalytic. Inordertoshowthefeasibilityoftheprocedure,wewillapply ittoafewsamplecasesinwhichthiscorankcanbedeterminedanalytically(averticaldisk with

various symmetrygroups) andto amorecomplexcasewhere theanalysis hastobecarried over

usingsymbolicmanipulationsoftware(aheavyballwhichrollsonasurfaceofrevolution).

ThenumberoflocalweaklyNoetherianconstantsofmotiongivesofcourseonlyanupperbound

onthenumberofgloballydenedconstantsofmotion,whicharethosesignicantforthedynamics.

Nevertheless,knowingthisboundcangivesomeinformationsonthesystem,butalso,canconrm

whether all constantsof motion with this propertyhave been determined. We will analyze this

situationontheexamples.

Ourstudy usesin anessentialwaythe Hamiltonianformulation ofnonholonomic mechanics,

whichisshortlyreviewedinSection2. InSection3weintroducethenotionofweaklyNoetherian

constants of motion and we shortly review the case of gaugemomenta. After this background

material,inSection4werelatetheexistenceoflocalweaklyNoetherianconstantsofmotiontothe

corankof the distribution

∆

∞

, and in Section 5wegiveexplicit expressionsfor thedistribution

∆

in the caseof a freeand properlifted action. Sections 6and 7 are devoted to theexamples. A section of conclusions and perspectives follows; in particular, weoutline there the use of our

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2 Nonholomic systems

A.Lagrangian formulation. Asalreadymentioned,weshallconsideronlythecaseofsystems

subjecttolinearnoholonomicconstraints.Forgeneralreferences,see[34,12,14,7,9]andreferences

therein.

Asastartingpoint,consideraholonomicmechanicalsystemwith

n

dimensionalconguration manifold

Q

and kinetic energy

T (v

q

) =

1

2 a

q

(v

q

, v

q

)

, where

a

is a Riemannian metric on

T Q

, subjecttoforcesdescribedbyapotentialenergywhichistheliftofafunction

V

on

Q

. Then,the Lagrangianoftheholonomicsystemis

L = T − V ◦ π

Q

,where

π

Q

: T Q → Q

isthetangentbundle projection.

Forgreaterclarity,weresorttoacoordinatedescriptionwheneverappropriate. Indoingso,we

denote by

(q, ˙q) ∈ R

n

_{× R}

n

(local) bundle coordinates on

T Q

and byadotthe inner productin

R

n

andweusethesymbol

loc

=

to stressthat therighthandsideofan equalityisanexpression foralocal representativeofthelefthandside. Thus, thekineticenergyislocallywritten

T (q, ˙q)

loc

=

1

2 ˙q · A(q) ˙q

forasymmetricandpositivedenite

n × n

matrix

A(q)

.

A linear nonholonomic constraintis given by anonintegrabledistribution

D

on

Q

, that we assumetohaveconstantrank

r

,

2 ≤ r < n

. Thedistribution

D

iscalledtheconstraintdistribution and can belocally dened asthe kernelof

k := n − r

linearly independent dierential 1forms

τ

1 , . . . , τ

k

on

Q

,sothat itsbersare

D

q

= ker

τ

1 (q), . . . , τ

k

(q)

.

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Inbundlecoordinates

(q, ˙q)

in

T Q

,thebersof

D

canbewritten as

D

q

loc

= { ˙q ∈ R

n

: S(q) ˙q = 0}

for a

k × n

matrix

S(q)

with rank

k

, which depends smoothlyon

q

. Theconstraintdistribution canalsobethoughtofasasubmanifold

D

of

T Q

ofdimension

2n − k

,whichinbundlecoordinates isgivenby

D

loc

= {(q, ˙q) ∈ R

2n

_{: q ∈ Q , ˙q ∈ D}

q

}

andwillbecalledthe(Lagrangian)constraintmanifold.

D'Alembert'sprincipleassumesthatthereactionforceexertedbythenonholonomicconstraint

annihilates the bers of (an appropriatejet extension of) thedistribution

D

and leads to a dy-namicalsystemonthesubmanifold

D

,whichisdescribedbyLagrangeequationswithmultipliers. Eliminatingthemultipliers givesthereactionforceasafunction ofthekinematical state

v

q

∈ D

andleadstoavectoreld ontheconstraintmanifold

D

,whichgivestheequationsofmotion. In bundlecoordinates,theequationsofmotionhavetheform

d

dt

∂L

∂ ˙q

−

∂L

∂q

= R

(2)

where

R = R(q, ˙q)

isthereactionforceexertedbythenonholonomicconstraint.

The expression of

R

in bundle coordinates can be found, e.g., in [1, 23, 6]. We recall this expressionbecausewewilluseitintheexamplesofSection6. Let

Π(q)

bethe

A(q)

−1

orthogonal

projectorontotheorthogonalsubspaceof

D

q

= ker S(q)

,that is

(4)

anddenethevectors

β(q, ˙q) ∈ R

n

,

V

0 _{(q) ∈ R}

n

and

γ(q, ˙q) ∈ R

k

ashavingcomponents

β

i

=

X

j,h

∂A

ij

∂q

h

−

1

2 ∂A

jh

∂q

i

˙q

j

˙q

h

,

V

i

0 =

∂V

∂q

i

,

γ

a

=

X

j,h

∂S

aj

∂q

h

˙q

j

˙q

h

(

i, j, h = 1, . . . , n

,

a = 1, . . . , k

). Then,

R = Πβ − S

T

SA

−1

S

T

−1

γ + Π V

0 .

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B. Local Hamiltonian formulation. We passnow to the Hamiltonian formulation of

non-holonomicallyconstraintsystems,see[40,4,32]forgeneralreferences. Webeginbygivingalocal

description of this formulation, in Darboux coordinates

(q, p)

on

T

∗

_Q

. We will use such local

formulationforthecomputationofthedistribution

∆

inSection5andintheexamplesofSections 6and 7.

TheLegendre transformation

Λ : T Q → T

∗

_Q

relativeto theLagrangian

L = T − V ◦ π

Q

isa dieomorphism andthe image

M := Λ(D)

of thesubmanifold

D

of

T Q

is asubmanifold (anda subbundle)of

T

∗

_Q

ofdimension

2n−k

,whichwillbecalledthe(Hamiltonian)constraintmanifold. InDarbouxcoordinates

Λ(q, ˙q)

loc

=

(q, A(q) ˙q)

and

M

loc

= {(q, p) ∈ R

2n

: q ∈ Q , A(q)

−1

p ∈ D

q

} .

The equations of motion of the nonholonomicsystem are Hamilton equations with the reaction

force. TheHamiltonian is

H = T ◦ Λ

−1

_{+ V ◦ π}

∗

Q

,where

π

∗

Q

: T

∗

Q → Q

is thecotangentbundle projection. InDarbouxcoordinates,

H(q, p)

loc

=

1 ₂

p · A(q)

−1

_{p + V (q)}

andtheequationsofmotion

are

˙q =

∂H

∂p

(q, p) ,

˙p = −

∂H

∂q

(q, p) + R q, A(q)

−1

_p

_.

Theseequationsarethelocalrepresentativeofadynamicalsystemontheconstraintmanifold

M

, thatwecallthe(Hamiltonian)nonholonomicsystem

(H, M )

.

Notethat it ispossibleto express theconstraintmanifold

M

alsoin terms ofthe 1forms

τ

. In fact, the submanifold

D

of

T Q

can be (locally) described as the zerolevel set of the map

˜

τ = (˜

τ

1 , . . . , ˜

τ

k

) : T Q → R

k

,where

τ

˜

i

: T Q → R

is givenby

˜

τ

i

(v

q

) := hτ

i

(q), v

q

i

for all

v

q

∈ D

q

,

q ∈ Q

. Therefore

M = ˆ

τ

−1

(0)

for

ˆ

τ (q, p) := (˜

τ ◦ Λ

−1

)(q, p)

loc

=

S(q)A(q)

−1

p ,

see[39].

C. Global Hamiltonian formulation. A local,coordinate descriptionwill not be sucient

forouranalysisand weneedageometric formulationof theequations ofmotion. Inviewofthis,

wepreliminarilyrecall somefacts from symplecticgeometry;seee.g. [31, 35]for detailsonthese

topics.

Let

ω

bethecanonicalsymplecticformof

T

∗

_Q

. Asusual,

X

F

denotestheHamiltonianvector eld of asmoothfunction

F

on

T

∗

_Q

, that is,

i

X

F

ω = −dF

. Similarly, the Hamiltonian vector eld

X

τ

ofaclosed 1form

τ

isdened by

i

X

τ

ω = −τ

. If

E

m

isasubspaceofthetangentspace

T

m

(T

∗

Q)

,

m ∈ T

∗

_Q

,thenitssymplecticorthogonalis denedas

(5)

Recall that

(E

ω

m

)

ω

= E

m

and that

E

m

and

E

ω

m

havecomplementary dimensions. Thepolar ofa distribution

T

on

T

∗

_Q

isthedistribution

T

ω

on

T

∗

_Q

whosebersarethesymplecticorthogonals

tothebersof

T

.

TheHamiltonian formulationof nonholonomicmechanics requires the introduction of a

cer-tain distribution

D

on

T

∗

_Q

, whose polar is related to the reaction force, see [4, 32, 39, 13, 14].

Specically, at each point

m ∈ T

∗

_Q

, theber

D

m

⊆ T

m

(T

∗

_Q)

of

D

isthe preimageof theber

D

_π

_Q

_(m)

oftheconstraintdistribution

D

underthederivative

T π

∗

Q

ofthecotangentbundle projec-tion

π

∗

Q

: T

∗

Q → Q

. With referencetotherepresentation(1)of

D

,thebersof

D

arethus given by

D

m

= ker

(π

Q

∗

)

∗

τ

1 (m), . . . , (π

Q

∗

)

∗

τ

k

(m)

loc

=

(u

q

, u

p

) ∈ R

n

× R

n

: u

q

∈ D

π

Q

(m)

(4) forall

m ∈ T

∗

_Q

. If,asweassumein thiswork,

D

hasconstantrank

n − k

, then

D

hasconstant rank

2n − k

.

Thedistribution

D

isdenedonallof

T

∗

_Q

,but wewillneedonlyitsrestrictionto

M

. Onthe points

m ∈ M

, theber

D

m

is distinctfromthetangentspace

T

m

M

,eventhoughtheyhavethe samedimension. Infact,

T

m

M

isthe annihilatorof thedierentialsof the

k

functions

ˆ

τ

a

which, inDarbouxcoordinates,arethefunctions

[S(q)A(q)

−1

_p]

a

. Notealsothat, while

M

dependsboth on

D

and

Λ

,andthus onthekineticenergy

T

,

D

anditspolar

D

ω

dependonlyon

D

.

If, as we assume in this work, the constraints are linear and ideal, then the Hamiltonian

counterpartofthereactionforceisasection

R

of(therestrictionto

M

of)thepolardistribution

D

ω

,

seee.g. [4,32,39,13,14]. Thedistribution

D

ω

hasconstantrank

k

andislocallygeneratedbythe vectorelds

X

(π

∗

Q

)

∗

τ

1 , . . . , X

(π

∗

_Q

)

∗

τ

k

. Correspondingly,thepolardistribution

[T

m

M ]

ω

hasconstant

rank

k

,too,andislocally generatedby

X

τ

ˆ

₁

, . . . , X

τ

ˆ

k

.

Asprovenin[4,32]ateachpoint

m ∈ M

thesubspaces

T

m

M

and

D

ω

m

of

T

m

T

∗

_Q

aretransversal

andcomplementary,sothat

T

m

(T

∗

Q) = T

m

M ⊕ D

ω

m

∀ m ∈ M .

(5)

This implies that, at each point

m ∈ M

, there is a unique splitting

u = u

T M

_{+ u}

D

ω

of vectors

u ∈ T

m

(T

∗

Q)

,whereforeachsubspace

E

of

T

m

(T

∗

_Q)

,

u

E

_{∈ E}

. Correspondingly,anyvectoreld

X

on

T

∗

_Q

canbeuniquelydecomposed onthepointsof

M

as

X = X

T M

+ X

D

ω

(6) with

X

T M

asectionof

T M

and

X

D

ω

asectionof

D

ω

.

Splitting (6)isusedtowritegloballytheequationsofmotionofnonholonomicsystems[4,32].

Infact,aconstrainedmotion

t 7→ m

t

∈ M

satises

m = X

˙

H

(m) + R(m) = X

T M

H

(m) + X

D

ω

H

(m) +

R(m)

. Hence,

m ∈ T

˙

m

M

and

R ∈ D

ω

ifandonlyifthereactionforce satises

R = −X

D

ω

H |M

and

t 7→ m

t

satisestheequationofmotion

˙

m = X

T M

H

(m) ,

m ∈ M .

Thedynamicalsystemon

M

denedbythevectoreld

X

T M

H

isthe(Hamiltonian)nonholonomic system

(H, M )

wementionedbefore.

Splitting(5)playsacentralrolealsointhestudyofconstantsofmotion [18,39];wewillcome

backonthisinSection4.

Remark: For completeness, wemention heretwofacts that we will notuse. (1)In thecase

of linear constraints, the vector eld

X

T M

H

belongs to

D

as well and is therefore a section of the distribution

H

= T M ∩ D

along

M

, see [4, 39]. As proven in [4, 16], the symplectic form restrictedto

H

isnondegenerateand

T

m

(T

∗

_{Q) = H}

m

⊕ H

ω

m

for

m ∈ M

. Thesefactsplayacentral rolein thereductionprocedure developedin [4]and inthedistributionalHamiltonianaproach to

nonholonomicsystemsdevelopedin[17,15]. (2)Thenonholonomicdynamicsconservestheenergy,

thatis,

X

T M

(6)

3 Weakly Noetherian constants of motion

A. Denition. We introduce now the notion of weakly Noetherian constants of motion and

review the case of gauge momenta. From now on, it is tacitly understood that

(H, M )

stands foraHamiltoniannonholonomicsystemofthetypespeciedinSection2,whichcorrespondstoa

congurationmanifold

Q

,aconstraintdistribution

D

,akineticenergy

T

andapotentialenergy

V

. By a constant of motion of the nonholonomic system

(H, M )

we mean a smooth function

F : M → R

suchthat theLie derivative

X

T M

H

(F ) = 0

inall of

M

. Givenan action

Ψ

Q

of aLie

group

G

on

Q

,wedenoteby

Ψ

T

∗

_Q

itslifttothecotangentbundle

T

∗

_Q

.

DenitionFixthecongurationmanifold

Q

,theconstraintdistribution

D

on

Q

,anaction

Ψ

Q

of

aLiegroup

G

on

Q

,anda

Ψ

T

∗

_Q

invariantkineticenergy

T : T

∗

_{Q → R}

. Then,asmoothfunction

F : M → R

is said to be aweakly Noetherian constant ofmotion relativeto

(T, M, Ψ

Q

₎

if it is

a constant of motion of all nonholonomic systems

(T + V ◦ π

∗

Q

, M )

with

Ψ

Q

invariant potential

energy

V : Q → R

.

Therestrictionto lifted actionsinthis Denitionis notnecessary,but thisisthecasewewill

considerin thesequel.

Remark: NotionsofNoetherianandofweaklyNoetherianconstantsofmotionofnonholonomic

systemswere givenin [22] in thespecial caseoffunctions linearin themomenta. Thetreatment

therewasin theLagrangiancontextbut,sinceunder ourhypothesestheLegendretransformation

isadieomorphism,everythingtransferstotheHamiltoniancontext. However,thereisasubtlety

involvedinextendingthenotionofNoetherianitytononlinearfunctions,andthisextensioncannot

beachievedbyjustdroppingtheconstancyofthekineticenergy

T

inthepreviousDenition. The factisthat,inamechanicalcontext,onestartswithagivenLagrangianconstraintmanifold

D

and (evenassumingthatthisoperationcouldbedonewithoutchanging

D

,whichmightbeunrealistic) changing the kineticenergy changes the Legendre transformation

Λ

and hence the Hamiltonian constraintmanifold

M = Λ(D)

. Since

M

is thenaturaldomain of denition ofthe constantsof motion,itisnotcompletelyclearwhatshouldtheappropriatedenitionofNoetherianconstantof

motionbe,ifany. Ifoneconsidersonlyfunctionswhicharedenedinallof

T

∗

_Q

,then`horizontal

momenta' (in the sense of the next subsection) are obviously Noetherian [22]. Note that this

dicultycannot beovercomebysimplyworkingin theLagrangiancontext,wherethe constraint

manifold

D

isxed,becausechangingthekineticenergychangesthepullbackof

J

to

T Q

.

B.WeaklyNoetheriangaugemomenta. Inordertogivesomeperspectiveforthesubsequent

treatmentwereviewveryquicklythepropertiesofweakNoetherianityoftheclassofconstantsof

motionwhichhas been moreextensivelystudied, that formed bymomenta andgaugemomenta.

Formotivations,detailsandexamplessee[22,24]andreferencestherein.

Bya`linear'functionon

M

wemeantherestrictionto

M

ofafunction on

T

∗

_Q

whichislinear

inthemomenta,that is,which canbewrittenas

F = p · ξ

Q

|

M

withsomevectoreld

ξ

Q

on

Q

. Thevectoreld

ξ

Q

iscalledageneratorof

F

and,since

M ⊂ T

∗

_Q

,

itisnotunique(see[28,22, 23]forfurtherdetails). Fixnowanaction

Ψ

Q

ofaLiegroup

G

on

Q

andlet

G

denotethedistributionon

Q

whosebers

G

q

arethetangentspaces

T

q

O

q

totheorbits

O

q

of

Ψ

Q

. Alinearfunction on

M

iscalledagaugemomentumrelativetotheaction

Ψ

Q

ifithas

agenerator

ξ

Q

withthefollowingtwoproperties: (1)itisasectionof

G

,and(2)itscotangentlift

ξ

T

∗

_Q

innitesimallypreservestheHamiltonianin

M

,namely

ξ

T

∗

_Q

(H)|

M

= 0

. Anysuchgenerator is called a gauge simmetry. (Note that agauge momentum may have as well generators which

do notsatisfy either condition and arenot gaugesymmetries). If the gaugesymmetry

ξ

Q

is an

innitesimalgeneratoroftheaction

Ψ

Q

(7)

of themomentum map of thelifted action

Ψ

T

∗

_Q

and will be called a momentum. Therefore, in

the sequelwe referonly to gaugemomenta. A gaugemomentum is said to be horizontal if it is

generated by ahorizontal gaugesymmetry, that is, a gaugesymmetry which is asection of the

constraintdistribution

D

.

Agaugemomentumisaconstantofmotionofthenonholonomicsystem

(H, M )

ifandonlyifit isgeneratedbyagaugesymmetrywhichisasectionofacertaindistribution

R

◦

D,T,V

on

Q

[23,22]. This distribution is called the reactionannihilator distribution, depends on

D

,

T

and

V

and is an overdistribution of the constraint distibution

D

, that is, its bers contain those of

D

. This implies,in particular, theverywellknownfact that allhorizontal momentaand gaugemomenta

areconstantsofmotion[30,3,16,8,13,5,33,7,19,15]. Thisalsoimpliesthatagaugemomentum

isweaklyNoetherianifandonlyifitisgeneratedbyagaugesymmetry

ξ

Q

whichisasectionofthe

distribution

∩

V

R

◦

D,T,V

which isobtainedby taking theintersection ofallthe distributions

R

◦

D,T,V

corresponding to

Ψ

Q

invariantpotentialenergies

V

. Since

∩

V

R

◦

D,T,V

isanoverdistributionof

D

,thisimpliesinparticularthathorizontal momenta and horizontal gauge momenta are weakly Noetherian [22]. Moreover,if the action is transitive

on

Q

, or moregenerally if

G

is an overdistribution of

D

, that is,

G

q

⊇ D

q

for all

q ∈ Q

, then any conservedgaugemomentum is horizontal ([22], Proposition 4) and henceweaklyNotherian.

However,theclass ofweakly Noetherianmomenta and gaugemomenta mightbelargerthanthe

horizontalones,apossibilitywhichmightbeparticularlyimportantincasessuchasthegeneralized

Chaplyginsystems,forwhich

D

q

∩ G

q

istrivial.

Remark: Thenumberoffunctionallyindependenthorizontalgaugemomenta(or,equivalently,

offunctionallyindependentconstantsofmotionidentiedbythemomentum equation[22])isnot

boundedfromabovebytherankofthedistributionwithbers

G

q

∩ D

q

,unlessthisrankiseither zeroorone. Infact, giventwoormorevectorelds on

Q

, liftingtheir linearcombinations with coecient functions produce vectorelds on

T Q

which need notbelong to the span of the lifts of thegivenvectorelds. Fora(holonomic)exampleof atwodimensionalLie groupwith three

indipendenthorizontalgaugemomentaseeSection6of [22].

4 On the number of weakly Noetherian constants of motion.

A.First integrals ofdistributions. Asexplainedin theIntroduction,ouraimistodevelopa

methodtocomputeanupperboundonthenumberoffunctionallyindependentweaklyNoetherian

constants of motion. The key will be regarding these functions as `rst integrals' of a certain

distribution. Wethusbeginthisanalysisbyintroducingthenotionofrstintegralsofadistribution

andstudyingsomeoftheirproperties.

For all notions and properties relative to distributions, and in particular for the denition

of smooth and real analytic distributions, we refer to [31, 35] (which call however generalized

distributionwhatwecallheredistribution)andto[29,12].

Giventwodistributions

T

and

S

onamanifold

M

,wesaythat

T

S

, andwrite

T

⊇ S

,iftheirberssatisfy

T

m

⊇ S

m

forall

m ∈ M

. Weusethetermunder-distribution inasimilarway.

Wesaythatasmoothfunction

F : M → R

isanrstintegralofadistribution

T

onamanifold

M

ifthedistribution withbers

ker dF (m)

T

,that is,

ker dF (m) ⊇ T

m

∀ m ∈ M .

(7)

Alocal rstintegral of

T

isanrstintegraloftherestrictionof

T

tosomeopensubsetof

M

. Thenotionofrstintegralsofadistributiongeneralizesinanobviouswaythatofrstintegrals

(8)

we have not been able to nd any reference to it in the literature; therefore, we collect herea

few properties of these objects. Roughlyspeaking, therelevant fact is that the rst integralsof

a distributionare constant on the (StefanSussmann) orbitsof the distribution, which, for non

integrable distributions, have dimensions larger than the rank of the distribution. Thus, rst

integralsofadistributionarerstintegralsofthefoliationthatitgenerates,andthedimensionof

theorbitsputsaboundonthenumberoffunctionallyindependentrstintegralswhichisstricter

thanthatgivenbytherankofthedistribution. Wegivenowaformal,andcomputationallymore

convenient,`innitesimal'statementofthis fact.

Preliminarily we recall that, given a smooth distribution

T

there exists a unique smooth overdistribution

T

∞

of

T

which is involutiveand isminimal among all smoothinvolutiveover distribution of

T

. This distribution is called the involutive closure of

T

. If

T

is real analytic, then

T

∞

is real analytic and there is astandardway to compute

T

∞

, which isbased on taking

commutators,seeSection 4.C.Forsomedetails,see[29,12] andreferencestherein.

Proposition 1. Let

T

beasmoothdistributionon

M

and

U ⊂ M

anopenset. Asmoothfunction

F : U → R

isarst integral of

T

|

U

if andonlyif itisarstintegral of

T

∞

_|

U

.

Proof Weusethefollowingnotation: if

D

isa(notnecessarilysmooth)distribution,then

smt(D)

denotesthelargestsmoothunderdistributionof

D

[29].

Preliminarily, we prove that

(T|

U

)

∞

_{= T}

∞

_|

U

. One inclusion is obvious:

T

∞

_|

U

is a smooth involutiveoverdistributionof

T

|

U

and hence is an overdistribution of

(T|

U

)

∞

. Viceversa, let

B

bethedistribution on

M

which coincides with

(T|

U

)

∞

on

U

and hasbers

T

m

M

for

m /

∈ U

. Therefore

smt(B)|

U

= (T|

U

)

∞

because the smoothing operationdoesnot change the bersof a

smoothdistribution. Ontheotherhand,

smt(B)

is asmoothoverdistributionof

T

which isalso involutive. Hence,

smt(B) ⊇ T

∞

and

(T|

U

)

∞

_{= smt(B)|}

U

⊇ T

∞

|

U

. Wethus write

T

|

∞

U

for

(T

U

)

∞

_{= T}

∞

_|

U

. If

F

isarstintegralof

T

|

∞

U

thenitis arstintegral of

T

|

U

⊆ T|

∞

U

. Conversely, assume that

F

is arst integralof

T

|

U

, that is,

ker dF ⊇ T|

U

. The distribution

ker dF

neednotbesmooth. Nevertheless,since

T

|

U

issmooth,

ker dF ⊇ smt(ker dF ) ⊇

T

_|

U

. Since thecommutatorof anytwosmoothsections of

ker dF

is asmoothsection of

ker dF

, thedistribution

smt(ker dF )

isinvolutive. That

ker dF ⊇ T|

∞

U

followsnowfrom theminimalityof

T

_|

∞

U

amongthesmoothinvolutiveoverdistributionsof

T

|

U

,whichimplies

smt(ker dF ) ⊇ T|

∞

U

. Foreach

m ∈ M

,dene

c

m

(T

∞

) := corank T

m

∞

.

If

T

isasmoothdistribution,sois

T

∞

andtheset

M

∞

reg

ofitsregularpointsisopenanddensein

M

[12] (apointissaidtoberegularifthethedistributionhasconstantrankinaneighbourhood ofit). Thus,Proposition1implies

Proposition 2. If

T

isa smooth distribution on amanifold

M

,then each point

m ∈ M

∞

reg

has aneighbourhood inwhich thereexist

c

m

(T

∞

₎

,but not

c

m

(T

∞

_{) + 1}

,functionally independent rst

integralsof

T

.

Proof Inaneighbourhood

U

ofaregularpoint

m

of

T

∞

thesmoothdistribution

T

|

∞

U

hasconstant rank and, being involutive, denes a smooth regular foliation of

U

with leaves of codimension

c

m

(T

∞

)

; in any set of local coordinates adapted to this foliation, thecoordinates transversal to theleavesarerstintegralsoftherestrictionof

T

∞

tothecoordinatedomain.

B. Onthe number of weakly Noetherian constants of motion. We consider now a

non-holonomicHamiltoniansystem

(H, M )

of thetypeconsidered so far and, asarst step, wegive acharacterizationofitsconstantsofmotionasrstintegralsofacertaindistribution onthe

con-straintmanifold

M

. (Fromnowonweshallconsideronlyobjectswhicharedenedon

M

andwe shallthereforeworkexclusivelyin

M

,notanymorein

T

∗

_Q

(9)

Wewilluse thefollowing notation. Given anitenumberof smoothvectorelds

X

1 , . . . , X

d

on

M

,

hX

1 , . . . , X

d

i

m

or

hX

1 (m), . . . , X

d

(m)i

denotes the subspace of

T

m

M

spanned by

X

1 (m), . . . , X

d

(m)

. Moreover,

hX

1 , . . . , X

d

i

denotesthedistributionwithbers

hX

1 , . . . , X

d

i

m

. Proposition 3. Asmoothfunction

F : M → R

isaconstantofmotionofanonholonomicsystem

(H, M )

ifandonly if

ker dF (m) ⊇ ker dH(m) ∩ D

m

ω

∩ T

m

M

∀ m ∈ M .

(8)

Proof Afunction

F

on

M

isaconstantofmotionof

(H, M )

ifandonlyif

hdF, X

T M

H

i = 0

atall pointsof

M

,that is,

ker dF (m) ⊇

X

H

T M

m

∀ m ∈ M .

Notethatthe

T M

component

u

T M

ofavector

u ∈ T

m

(T

∗

_Q)

isgivenby

(u + D

ω

m

) ∩ T

m

M

. Hence,

hX

T M

H

i

m

= (hX

H

i

m

+ D

ω

m

) ∩ T

m

M

. Recallingthat

X

H

issymplecticallyorthogonalto

ker dH

we thushave

hX

T M

H

i

m

= [(ker dH(m))

ω

+ D

ω

m

] ∩ T

m

M = (ker dH(m) ∩ D

m

)

ω

∩ T

m

M

.

Remarks: (i) Characterization (8) of constants of motion of nonholonomic systems is in a

sensedual tocharacterizationsgivenin [16,18,39]. Since(5)implies

T

m

(T

∗

_{Q) = (T}

m

M )

ω

⊕ D

m

forall

m ∈ M

,onthe pointsof

M

thereisasplitting

X = X

T M

ω

_{+ X}

D

ofvectoreldson

T

∗

_Q

whichisdual to (6). As provenin[39], afunction

F : M → R

is aconstantofmotionof

(H, M )

ifandonlyifforone,andthereforeany,extension

F

˜

of

F

o

M

,

X

D

˜

F

(H)

|M

= 0 .

(9)

Notethatthis conditionisequivalentto

X

D

˜

F

(m) ∈

ker dH(m) ∩ D

m

+ T

m

M

ω

∀ m ∈ M ,

whichcanbecomparedto (8). Areformulationofcondition(9) which, likecondition(8),avoids

theuseofextensionsoffunctionso

M

canbegivenintermsofdistributionalHamiltonianvector elds[15].

(ii) A function

F

is a constant of motion of a Hamiltonian vector eld

X

H

if and only if either

X

H

(F ) = 0

or,equivalently,

X

F

(H) = 0

. Condition (9) is manifestly an analogueof the latter condition. Instead, (8) is an analogueof condition

X

H

(F ) = 0

, that is,

ker dF ⊇ hX

H

i

. In fact, if there are no nonholonomic constraints, then

M = T

∗

_Q

,

D

= T (T

∗

_Q)

and therefore

ker dH ∩ D

ω

∩ T M = (ker dH)

ω

_{= hX}

H

i

.

Based onProposition 3,wecan nowgivethefollowingcharacterizationof weakly Noetherian

constantsofmotion:

Proposition 4. Fixthe congurationmanifold

Q

,the constraintdistribution

D

on

Q

,anaction

Ψ

Q

of aLie group

G

on

Q

,anda

Ψ

T

∗

_Q

invariant kineticenergy

T : T

∗

_{Q → R}

. Let

I

G

bethe set of all

Ψ

Q

invariantsmooth real functionson

Q

. For each

m ∈ M

dene

I

m

:=

\

V ∈I

G

ker dT (m) + d(V ◦ π

Q

∗

)(m)

(10)

∆

m

:= I

m

∩ D

m

ω

∩ T

m

M .

(11)

Then, a smooth function

F : M → R

is a weakly Noetherian constant of motion relative to

(T, M, Ψ

Q

₎

if andonlyif

(10)

Proof A function

F

is a weakly Noetherian constant of motion if and only if, at each point

m ∈ M

,inclusion(8)issatisedforall

H = T + V ◦ π

∗

Q

with

V ∈ I

G

. Omittingforshortnessthe indicationofthepoint

m

,thisisequivalentto

(ker dF )

ω

_{⊆ ker d(T + V ◦ π}

∗

Q

) ∩ D

+ T M

ω

forall

V ∈ I

G

,namely

(ker dF )

ω

_⊆

\

V ∈I

G

ker d(T + V ) ∩ D

+ T M

ω

_.

Sincethetransversalityof

D

ω

and

T M

impliesthatof

D

and

T M

ω

,thesubspace

T

V ∈I

G

ker d(T +

V ) ∩ D

+ T M

ω

equals

(I ∩ D) + T M

ω

.

Let

∆

bethedistributionon

M

withbers

∆

m

⊆ T

m

M

asin(12). Inviewof(7),Proposition4 meansthat,if

∆

issmooth,thenafunction

F : M → R

isaweaklyNoetherianconstantofmotionif andonlyifitisarstintegralofthedistribution

∆

. Therefore,if

c

m

(∆

∞

₎

denotesthecodimension

oftheber

∆

∞

m

in

T

m

M

,that is,

c

m

(∆

∞

) = dim M − rank ∆

∞

m

= (2n − k) − rank ∆

∞

m

,

then by Proposition 2 in a neighbourhood of each regular point

m

of

∆

∞

there exist

c

m

(∆

∞

₎

, but not

c

m

(∆

∞

_{) + 1}

, functionally independent weakly Noetherian constants of motion relative to

(T, M, Ψ

Q

₎

. From a dynamical point of view, the existence of local constants of motion of a

dynamical systemhasno particular signicancetherectication theorem ensures theexistence

of the maximal numberof them. Thus, this fact cannot be considered as anexistence result of

weaklyNoetherian constantsofmotion,butanupperbound ontheirnumber:

Theorem1. Under the hypotheses of Proposition 2, every setof functionally independent global

weakly Noetherian smoothconstants ofmotion relative to

(T, M, Ψ

Q

₎

containsatmost

c(∆

∞

_{) :=}

min

m∈M

c

m

(∆

∞

)

elements.

In practice, however, the estimate provided by Theorem 1 may be used as a guide for the

search ofasmanyweakly Noetherianconstantsofmotionaspossible,includinghorizontalgauge

momenta.

Of course, the eectiveness of this approach depends on the possibility of constructing the

distribution

∆

∞

,whichinturnrequirestheknowledgeof

∆

. Aswereviewinthenextsubsection, thereisastandardprocedurefortheconstructionof

∆

∞

ifthedistribution

∆

isreal analytic(see e.g. Chapter3of [12] fordetails and references). The constructionofthe distribution

∆

willbe doneinSection5undersuitableassumptionsontheactions

Ψ

Q

.

C.Constructionof

∆

∞

. Weassumenowthat,asithappensintypicalcases,thedistribution

∆

isrealanalytic. Aswehavealreadymentioned,thereastandardwayofconstructingtheinvolutive

closure of areal analytic distribution[12]. Since anyreal analytic distribution is locally nitely

generated,in anysuciently small open set

U ⊆ M

∞

reg

there are

r = rank ∆

independent vector elds

X

1 , . . . , X

d

such that

∆

1 _{:= ∆}

|U

hasbers

∆

1 m

=

X

1 , . . . , X

d

_m

,that is,

∆

1 = hX

1 , . . . , X

d

i .

Dene

∆

s+1

_{:= ∆}

s

_{∪ [∆}

s

_{, ∆}

s

_{] ,}

_{s = 1, . . . , n − 1 ,}

where

[∆

s

_{, ∆}

s

_]

isthedistributionwhosesectionsareallthecommutators

[η, ξ]

for

η, ξ

sectionsof

∆

s

,thatis,

∆

2 = hX

1 , . . . , [X

1 , X

2 ], . . .i

(11)

etc. Thus,

∆

∞

_{= ∆}

s

where

s

isthesmallestpositiveintegersuchthat

∆

s

isintegrableor,inother

words,thesmallestpositiveintegersuch that

∆

s

_{= ∆}

s+1

.

Thus, in order to apply the criterion of Theorem 1 to a specic nonholonomic system with

analyticdistribution

∆

,itsucestondasystemofgeneratorsof

∆

in aneighbourhoodofeach ofitsregularpointsandthencomputesucientlymanyLiebracketsofthesegenerators,checking

forlinearindependence.

This procedure canbeimplementedin twoways. Oneis ofcoursethat ofparameterizingthe

constraintmanifold

M

andwriting thegeneratorsof

∆

usingthe chosenlocal coordinates. The other is that of making all computations using extensions o

M

of the chosen local generators

X

1 , . . . , X

d

of

∆

. Infact,if

X

˜

1 , . . . , ˜

X

d

areextensionsof

X

1 , . . . , X

d

o

M

,then

[ ˜

X

a

, ˜

X

b

]

|M

= [X

a

, X

b

]

∀ a, b

for a known property of related vector elds, see e.g. [35]. We will use both methods in the

examplesbelow.

5 The distribution

∆

for free and proper actions

A. Determinationof

∆

. Thedetermination ofthedistribution

∆

requiresthedetermination ofthedistribution

I

asin(10). Thisis aneasy taskiftheactionis properandfree,andwelimit ourselvesto thiscase.

As above,wedenoteby

G

thedistributionon

Q

whosebers

G

q

are thetangentspaces

T

q

O

q

to the orbits

O

q

of the action

Ψ

Q

. Forour purposes it is sucient to determine

∆

in Darboux coordinates

(q, p)

. Thus, weidentify the tangentspaces

T

(q,p)

T

∗

_Q

with

R

2n

(or with

R

n

_{× R}

n

,

when wewantto stressthe individualroles ofthe

q

and

p

componentsof thetangentvectors), weidentifythespaces

G

q

= T

q

O

q

and

D

q

withsubspacesof

R

n

andweidentifythespaces

T

(q,p)

M

,

∆

(q,p)

and

I

(q,p)

with subspacesof

R

2n

. Inthesequel, thedot denotesthe scalarproduct in

R

n

and, if

E

is a subspace of

R

n

, then

E

⊥

denotes its orthogonal complement in

R

n

; if

v ∈ R

n

, then

v

⊥

stands for

hvi

⊥

. We will distinguishbetweenrow and column vectorsonly when some

ambiguitymightarise. Asbefore,wedenoteby

A(q)

thekineticmatrix,sothatthekineticenergy is

T (q, p) =

1

2 p · A(q)

−1

_p

.

Lemma 1. In addition to the assumptions of Proposition 4, assume that the action

Ψ

Q

is free

andproper. Then, ateachpoint

(q, p) ∈ M

,

I

_(q,p)

=

(v

q

, v

p

) ∈ R

n

× R

n

: v

q

∈ G

q

, v

p

∈ T

p

0 ⊥

−

v

q

· T

0 q

kT

0 p

k

2 T

p

0

(13) where

T

p

0 :=

∂T

∂p

(q, p) = A(q)

−1

_p

_and

_T

0 q

:=

∂T

∂q

(q, p) .

(14)

Proof Avector

v = (v

q

, v

p

) ∈ I

(q,p)

ifand onlyif

d(T + V )v = 0

forall

V ∈ I

G

, namely ifand onlyif

∂V

∂q

· v

q

= 0

forall

V ∈ I

G

and

T

0 q

· v

q

+ T

p

0 · v

p

= 0

. Iftheactionis freeandproper,then thecondition

∂V

∂q

· v

q

= 0

∀ V ∈ I

G

is equivalentto

v

q

∈ G

q

(see e.g. Theorem 2.5.10of [35]). If

p = 0

then

T

0 p

= T

q

0 = 0

and there arenoconditionson

v

p

; thisisconsistentwith (13)becauseinthiscase

T

0 p

⊥

= R

n

and

T

0 q

= 0

. If

p 6= 0

then

v

p

∈ T

0 p

⊥

− kT

0 p

k

−2

(v

q

· T

q

0 )T

p

0

.

(12)

Lemma 2. Under the hypotheses and with the notation of Lemma 1, at any point

(q, p) ∈ M

a vector

(u

q

, u

p

) ∈ R

n

_{× R}

n

belongsto

I

(q,p)

∩ D

(q,p)

ω

if andonlyif

u

q

∈ hT

p

0 i

and

u

p

∈ [D

q

∩ G

q

]

⊥

−

u

q

· T

p

0 kT

0 p

k

2 T

q

0

(15) with

T

0 p

and

T

0 q

asin (14).

Proof Formulas(4)and(13)imply

I

_(q,p)

_{∩ D}

_(q,p)

=

(v

q

, v

p

) ∈ R

n

× R

n

: v

q

∈ D

q

∩ G

q

, v

p

∈ T

p

0 ⊥

−

T

0 q

· v

q

kT

0 p

k

2 T

p

0 .

(16) Thus, a vector

(u

q

, u

p

) ∈ R

n

× R

n

is symplectically orthogonal to

I

(q,p)

∩ D

(q,p)

if and only if

u

q

· v

p

= u

p

· v

q

forall

v

q

∈ D

q

∩ G

q

andall

v

p

∈ T

0 p

⊥

− (T

0 q

· v

q

)kT

p

0 k

−2

T

p

0

,that is,ifandonlyif

u

q

· T

p

0 ⊥

−

v

q

· T

0 q

kT

0 p

k

2 T

p

0 = u

p

· v

q

∀v

q

∈ D

q

∩ G

q

.

(17)

When

v

q

= 0

thisequalityreduces to

u

q

· T

0 p

⊥

= 0

,whichgives

u

q

= λT

0 p

forsome

λ ∈ R

. Forthis valueof

u

q

,(17)becomes

(λT

0 q

+ u

p

) · v

q

= 0

forall

v

q

∈ D

q

∩ G

q

. Thisshowsthat

[I

(q,p)

∩ D

(q,p)

]

ω

= {(u

q

, u

p

) : u

q

= λT

p

0 , u

p

∈ [D

q

∩ G

q

]

⊥

− λT

q

0 , λ ∈ R} .

Expressing

λ

in termsof

u

q

and

T

0 p

leadsto(15).

Wecannowcompletethecharacterizationof

∆

. Notethat,since

A

isaninvertible

n×n

matrix and

S

isa

k × n

matrixwithrank

k

,

SA

−1

_S

T

isaninvertible

k × k

matrix. Notealsothat, ifthe vectorsubspace

D

q

∩ G

q

⊥

of

R

n

is

k

dimensionaland hasabasisformedbyvectors

w

1 , . . . , w

k

of

R

n

, then the vector subspace

I

_(q,p)

_{∩ D}

_(q,p)

ω

of

R

2n

is

(k + 1)

dimensionaland has abasis formedbythevectors

(0, w

1 ), . . . , (0, w

k

), (T

0 p

, −T

q

0 )

of

R

2n

.

Proposition 5. UnderthehypothesesandwiththenotationofLemma1,atanypoint

(q, p) ∈ M

avector

(u

q

, u

p

) ∈ R

n

_{× R}

n

belongsto

∆

(q,p)

if andonlyif itsatises(15) and

u

p

∈ A(q)D

q

− S(q)

T

S(q)A(q)

−1

S(q)

T

−1

K(q, p)u

q

(18)

where

K(q, p)

isthe matrixwith components

K(q, p)

ai

=

∂

∂q

i

S(q)A(q)

−1

p

_a

,

a = 1, . . . , k ,

i = 1, . . . , n .

Proof Recallthat

∆

m

= [I

m

∩ D

m

]

ω

_{∩ T}

m

M

. ByLemma2,

(u

q

, u

p

) ∈ [I

m

∩ D

m

]

ω

ifandonlyif

u

q

and

u

p

satisfy(15). Since

M

is(locally)thezerolevelsetofthe

k

functions

ˆ

τ

a

= (S(q)A(q)

−1

_p)

a

, avector

(u

q

, u

p

) ∈ R

n

isin

T

(q,p)

M

ifandonlyif

u

q

· ∂ ˆ

τ

a

∂q

(q, p) + u

p

· ∂ ˆ

τ

a

∂p

(q, p) = 0

∀a = 1, . . . , k ,

thatis,ifandonlyif

S(q)A(q)

−1

u

p

= −K(q, p)u

q

.

(19)

Given that

D

q

= ker S(q)

it is immediate to verify that this condition is equivalent to

u

p

∈

AD − S

T

(SA

−1

S

T

)

−1

Ku

q

.

(13)

Corollary 1. If

G

actsfreely andproperly on

Q

and

G

isan overdistributionof

D

,then

∆

has rankoneatallpointsof

M

,except onthe sumbmanifold

p = 0

of

M

whereithas rankzero.

Proof We shall provea little bit more, that is, that ifthe action is freeand proper, then the

distribution

∆

satises

dim ∆

(q,p)

= 1 + n − k − dim(G

q

∩ D

q

if p 6= 0

dim ∆

(q,0)

= n − k − dim(G

q

∩ D

q

.

This implies the statement because, if

G

q

∩ D

q

= D

q

, then

dim(G

q

∩ D

q

) = n − k

. Let

m =

(q, p)

. Notethat

dim ∆

m

= 2n − dim ∆

ω

m

. From (11)andthe transversalityof

T

m

M

and

D

ω

m

it followsthat

∆

ω

m

= I

m

∩ D

m

⊕ (T

m

M )

ω

. Hence

dim ∆

ω

m

= dim(I

m

∩ D

m

) + dim[T

m

M ]

ω

, where

dim[T

m

M ]

ω

= k

. From(16)itfollowsthat,if

p 6= 0

,then

dim(I

m

∩ D

m

) = dim (G

q

∩ D

q

) + (n − 1)

because

T

0 p

⊥

is a subspace of

R

n

of dimension

n − 1

. If instead

p = 0

, then

T

0 p

⊥

= R

n

and

dim(I

m

∩ D

m

) = dim (G

q

∩ D

q

) + n

. Hence,iftheaction

Ψ

Q

on

Q

is freeandproper,and ifthe grouporbitsaresucientlylarge, sothat

G

⊇ D

,thenthedistribution

∆

hasrankoneatallpointsexceptat

p = 0

,whereitsrank iszero. Thisimpliesthat

∆

isintegrableand

∆

∞

_{= ∆}

. Therefore,itfollowsfromthestatement

at the beginning of the proofof Theorem 1 that, in this case,givenany

Ψ

T

∗

_Q

invariantkinetic

energy

T

on

T

∗

_Q

,inaneighbourhoodofeachpointof

M

thereisthemaximumnumber

2n − k − 1

offunctionallyindependentlocalweaklyNoetherianconstantsofmotion. Thus,inthiscase,which

includesthecaseoftransitiveactions,ourmethoddoesnotgiveanynewinformationwithrespect

to therectication theorem, but thefact that the local constantsof motioncanbechosen tobe

weaklyNoetherian. Thisis asortofweaklyNoetherian versionoftherecticationtheorem.

If

G

isnotanoverdistributionof

D

,instead,thedistribution

∆

hasrankgreaterthanoneand maybenotintegrable.Thisputssomelimitationsonthenumberof(evenlocal)weaklyNoetherian

constantsofmotion.

Next,wehave:

Corollary 2. Given

Q

,

M

and

T

,consider twofree andproper actions

Ψ

Q

1

and

Ψ

Q

2

of twoLie groups

G

1

and

G

2

on

Q

, whose tangent lifts preserve

T

. If the orbits of the

Ψ

Q

1

action contain those of the

Ψ

Q

2

action, then any weakly Noetherian constant of motion relative to

(T, M, Ψ

Q

2 )

is also aweakly Noetherianconstant ofmotion relativeto

(T, M, Ψ

Q

1 )

.

Proof Accordingto (13)and (18),thefact that avectortangentto

M

isin

∆

depends onthe groupactiononlythroughthetangentspaceto thegrouporbits.

Inparticular, iftwodierent groupactionshavethe sameorbitsin

Q

(but possibly dierent orbitsin

T Q

),thentheyhavethesameweaklyNoetherianconstantsofmotion.

6 Example: the rolling vertical disk.

A.Thesystem. Asarsttestcaseforthemethod,andasanillustrationofit,weconsiderhere

thesystemformedbyadiskwhichisconstrainedtorollwithoutslippingonahorizontalplane,and

tostandvertically. Thisisawellknownsystem,whichhasbeenconsiderede.g. in [16,8,13,23].

Weshallconsidervariousclassesofforcesactingonthedisk,withvarioussymmetrygroups,soas

toshowhowchangingthesymmetrygroupchanges thedimensionof

∆

∞

andthusthebound on

(14)

constantsof motion,whichin theconsideredcasescanbedeterminedbyanalyzingtheequations

ofmotion.

The holonomic system has conguration manifold

Q = R

2 _{× S}

1 _{× S}

1 _{3 (x, y, θ, ϕ)}

, where

(x, y) ∈ R

2

are Cartesiancoordinatesof the point of contact,

ϕ

is theangle between the

x

axis andtheprojectionofthedisk ontheplane,and

θ

isthe anglebetweenaxedradiusof thedisk and the vertical. In orderto simplify thenotation, weassume that massand radius of the disk

are both unitary. The nonholonomic noslipping constraint imposes that the point of the disk

in contactwith the plane has zero velocity, that is,

˙x = ˙θ cos ϕ

and

˙y = ˙θ sin ϕ

. The ranktwo constraintdistributionhasbers

D

(x,y,θ,ϕ)

=

cos ϕ ∂

x

+ sin ϕ ∂

y

+ ∂

θ

, ∂

ϕ

thatis,

ker S

(x,y,θ,ϕ)

for

S

(x,y,θ,ϕ)

=

cos ϕ

sin ϕ

−1 0

− sin ϕ cos ϕ

0

0 .

The(Lagrangian) kinetic energy is

T =

1

2 ˙x

2 _{+ ˙y}

2 ₊

1

2 I ˙θ

2 ₊

1

2 J ˙

ϕ

2

, where

I

and

J

are the appropriatemomentsofinertia. Ifwewrite

h = 1/I

,

k = 1/J

andif

V (x, y, ϕ, θ)

is thepotential energyoftheforcesactingonthedisk,thentheHamiltonianis

H(x, y, ϕ, θ, p

x

, p

y

, p

θ

, p

ϕ

) =

p

2 x

+ p

2 y

2 +

h

2 p

2 θ

+

k

2 p

2 ϕ

+ V (x, y, ϕ, θ) .

Theconstraintmanifold

M =

(x, y, θ, ϕ, p

x

, p

y

, p

θ

, p

ϕ

) : p

x

= kp

θ

cos ϕ , p

y

= kp

θ

sin ϕ

is sixdimensional. It is dieomorphic to

R

4 _{× T}

2

and can be globally parameterized with

(x, y, θ, ϕ, p

θ

, p

ϕ

)

. Usingthesecoordinates,theequationsofmotion,whichcanbecomputedfrom (2)and(3),are

˙x = hp

θ

cos ϕ ,

˙y = hp

θ

sin ϕ ,

˙θ = hp

θ

,

ϕ = kp

˙

ϕ

,

(20)

˙p

θ

= −

1 1 + h

V

0 θ

+ V

x

0 cos ϕ + V

y

0 sin ϕ

,

˙p

ϕ

= V

ϕ

0 .

Theseequationsaresimpleenoughtoallowustointerprettheresultsoftheforthcominganalysis,

whereincreasinglysmallersymmetrygroupsareconsidered. Forsimplicityofexposition,weclassify

thesegroupsviathecorrespondingclassesofinvariantpotentials.

B. No forces. Ifweallowonlyconstantexternalpotentials,thenthesystemisinvariantunder

thegroup

G = R

2 _{× T}

2

,whichactson

Q

bytranslationsalongthecoordinates

(x, y, θ, ϕ)

. Sincethisactionistransitive,ourmethodgivestheexistenceofsetsofveweaklyNoetherian

local constants of motion in a neighbourhood of any point of

M

, see Corollary 1. As we now show,onlyfouroftheseconstantsofmotionaregloballydened;twoofthemcanbechosentobe

horizontalmomentaandtheothershorizontalgaugemomenta.

Inordertosee thisobservethat,forconstantpotentials,theequationsofmotion(20)become

˙x = hp

θ

cos ϕ ,

˙y = hp

θ

sin ϕ ,

˙θ = hp

θ

,

ϕ = kp

˙

ϕ

,

˙p

θ

= 0 ,

˙p

ϕ

= 0 .

(21) Thus

p

θ

and

p

ϕ

are constants of motion, and they are obviously horizontal momenta. Observe now thateachsubsystem

(θ, p

θ

)

and

(ϕ, p

ϕ

)

performsuniform rotationson

S

1 _{× R}

,with angular

frequencies

hp

θ

and

kp

ϕ

respectively. Since

ϕ

grows linearly in time and

p

θ

is constant, inte-gratingthersttwoequations(21)showsthattheprojectionof themotionin the

(x, y)

plane is

(15)

genericallya uniformcircular motion, with frequency

kp

ϕ

. (The motion is, exceptionally, linear if

p

ϕ

= 0

). Thus,motionsofthesystemare genericallyquasiperiodicontori ofdimensionthree, withfrequencies

hp

θ

, kp

ϕ

, kp

ϕ

. Sincetwoofthefrequenciesareequaltheclosureofalltheseorbits iscontainedintoriofdimensiontwo. Butsincetheratio

hp

θ

/(kp

ϕ

)

variescontinuously,thesetof orbitswhoseclosureis atwodimensionaltorusforms adenseset. Therefore,thereare notmore

than

6 − 2 = 4

independentconstantsofmotionwhich aredenedinopeninvariantsets.

Asimplecomputation showsthat twootherglobally dened constantsof motionare

kxp

ϕ

−

hp

θ

sin ϕ

and

kyp

ϕ

+ hp

θ

cos ϕ

. They are horizontal gauge momenta because, if the action is transitive,thenanyglobalconstantofmotionwhichislinearinthevelocitiesisahorizontalgauge

momentum,seeProposition4of [22].

Remark: The fthlocal integralgivesthe slope ofthe orbitsonthe twodimensionaltori.

It can be taken e.g. as

hp

θ

ϕ − kp

ϕ

θ

and it is not globally dened on the tori with irrational

kp

θ

/(hp

ϕ

)

. Hence,itis notagaugemomentum eventhoughit islocally alinearcombination of twohorizontalmomenta.

On the number of weakly Noetherian constants of motion of nonholonomic systems

∗

†

‡

•

R

◦

•

∗

†

‡

∆

∞

∆

∆

∆

∞

∆

∆

∞

∆

n

Q

T (v

q

) =

1

2

a

q

(v

q

, v

q

)

a

T Q

V

Q

L = T − V ◦ π

Q

π

Q

: T Q → Q

(q, ˙q) ∈ R

n

× R

n

T Q

R

n

loc

=

T (q, ˙q)

loc

=

1

2

˙q · A(q) ˙q

n × n

A(q)

D

Q

r

2 ≤ r < n

D

k := n − r

τ

1

, . . . , τ

k

Q

D

q

= ker



τ

1

(q), . . . , τ

k

_{× R}

_{: q ∈ Q , ˙q ∈ D}

_{(q) ∈ R}