The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories

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Contents lists available atScienceDirect

Physics

Letters

A

www.elsevier.com/locate/pla

The

Perlick

system

type

I:

From

the

algebra

of

symmetries

to

the

geometry

of

the

trajectories

¸S. Kuru

a

,

J. Negro

b

,

,

O. Ragnisco

c

aDepartmentofPhysics,FacultyofScience,AnkaraUniversity,06100Ankara,Turkey

bDepartamentodeFísicaTeórica,AtómicayÓptica,UniversidaddeValladolid,47011Valladolid,Spain cInstitutoNazionalediFisicaNucleare,SezionediRoma3,ViadellaVascaNavale84,00146Roma,Italy

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received13June2017 Accepted17August2017 Availableonline26August2017 CommunicatedbyA.P.Fordy

Keywords: Bertrandspaces Superintegrablesystems Factorizationmethod Constantsofmotion

In this paper, we investigate the main algebraic properties ofthe maximally superintegrable system knownas“PerlicksystemtypeI”.Allpossiblevaluesoftherelevantparameters,Kandβ,areconsidered. Inparticular, dependingonthesign oftheparameter K enteringinthemetrics,themotionwilltake placeoncompactornoncompactRiemannianmanifolds.Toperformouranalysiswefollowaclassical variantofthesocalledfactorizationmethod.Accordingly,wederivethefullsetofconstantsofmotion andconstructtheirPoissonalgebra.Asitisexpectedformaximallysuperintegrablesystems,thealgebraic structurewillactuallyshedlightalsoonthegeometricfeaturesofthetrajectories,thatwillbedepicted fordifferentvalues oftheinitialdata andoftheparameters.Especially,thecrucialrole playedbythe rationalparameterβwillbeseen“inaction”.

©2017ElsevierB.V.Allrightsreserved.

1. Introduction

ThePerlicksystems typeIandII havebeenstudiedina large numberof papers, where severalcrucial features havebeen dis-covered(seeforinstance[1–10]).Inparticular,letusremindthat inhis original paper [1], V. Perlick constructedthe most general class of three-dimensional spherically symmetric lagrangian sys-temscomplyingwiththerequirementsofthecelebratedBertrand’s Theorem[11],namelythefactthattheypossessstablecircular or-bits andall boundedtrajectories are closed.Ofcourse he hadto abandonthe Euclideanspaceframeworkinfavor ofmore general conformallyflatmanifolds.Inthisway,heunveiledthedeep con-nection existingbetween themetrics characterizing the manifold and the corresponding integrable potentials. He classified these generalizedBertrandsystemsintwo multiparametricfamilies, de-finedaccordingtotheirEuclideanlimit.Family Iistheoneyielding the Kepler–Coulomb system and Family II is the one leading to theradialharmonicoscillator[12–14].Inref.[3]bymeansofthe coalgebra approach the extension to hyperspherically symmetric systemsinanydimensionwasperformed,andanintrinsic charac-terizationofthetwofamilieswasproposed.FamilyIwasdenoted as“intrinsicKepler–Coulomb” because,up to additive and

multi-*

Correspondingauthor.

E-mailaddresses:kuru@science.ankara.edu.tr( ¸S. Kuru),jnegro@fta.uva.es

(J. Negro),oragnisco@gmail.com(O. Ragnisco).

plicativeconstants(onecouldsay,“up toaffinetransformations”), the pertaining class of potentials were discovered to be just the Green’sfunctionsofthecorrespondingLaplace–Beltramioperators. Analogously, the potentials comprised in Family II, again up to affinetransformations,wereidentified asthe“inverse-squared”of those belongingtoFamily Iandthesehavebeen called“intrinsic oscillator”.Inasubsequentpaper[4]thesameauthorsgavea rig-orousproofofthepreviouslyobtainedresults,includingthatofthe periodicityoftheboundedmotions,andprovidedaclosed expres-sion forthe corresponding trajectories. Thus, Perlick’s systems of type IandIIare themostgeneralclassesof“maximally superin-tegrable”lagrangian(andconsequentlyHamiltonian)systemswith (hyper-)sphericalsymmetry.

Tofurther clarify our previous statements,let us recallthat a classical Hamiltonian system with n degrees of freedom is said to be integrable if there are n functionally independent globally defined andsingle-valued integralsof motion ininvolution fora given Poissonbracket. Ifthere exist k, with0

<

k

n

1, addi-tionalintegrals(theymustbefunctionallyindependent,butnotall ofthemininvolution),itiscalledsuperintegrable.Whenk

=

n

1, thesystemismaximallysuperintegrable.Inthiscase, allbounded trajectories are closed and the motion is periodic [15–17]. The constants of motion of maximally superintegrable systems close an algebra andcan be usedto determinethe trajectoriesinpure algebraic way. Kepler–Coulomb and (isotropic) harmonic oscilla-torsystemsareexamplesofsphericallysymmetricandmaximally superintegrable systems in an n dimensional Euclidean space. In

http://dx.doi.org/10.1016/j.physleta.2017.08.042

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dealingherewithamaximallysuperintegrablesystemsuchas Per-licksystemoftype I, ratherthanrelying onan analyticapproach asin [4], we will follow an algebraic one, that can be generally called“factorizationapproach”or“factorizationmethod”.Through the factorization method the algebra of the integrals of motions canbeexplicitlyconstructed,anditsfeatureswillpavethewayto identifyingrelevantgeometricpropertiessuch astheshapeofthe closedorbits.Wewishtoremarkthatthefactorizationmethod ap-pliesbothtoclassical andquantum systems.Themethodisquite simplein yielding theconstants ofmotion (symmetries)and un-veilingtheir underlyingalgebra [18–25].We shouldmentionthat othermethodshavebeensuccessfullyappliedtofindtheconstants ofmotionofsuperintegrablesystems.Oneofthemisbasedonthe Hamilton–Jacobi equation,asit isshownin manyreferences[14, 26,27].Anotherinterestingwaytolookforconstants ofmotionis themethodofcoalgebrasdevelopedin[28,29].Asystematic alge-braic methodhas alsobeen usedin other papers [30,31].In this work,we willperformasystematicinvestigationoftheconstants ofmotionoftheclassicalsuperintegrablethreedimensionalPerlick systemtypeIbyusingthefactorizationmethodinordertoderive thetrajectoriesinapurelyalgebraicway.

Thispaperisorganizedasfollows.Insection2,webrieflyrecall thebasicdefinitionofthePerlicksystemoftypeIwhichdepends essentially on two parameters

β

and K. We will introduce new variables more suitable to describe the compact ornon compact manifoldswherethissystemmaybedefined.Insection3,we con-struct the constants of motionby means of an extension of the factorizationmethodtoclassicalthreedimensionalsystems(itcan beapplied tohigherdimensionstoo). ThetrajectoriesofPerlick I systemarederived,discussedanddepicted,fordifferentvaluesof theinitialconditionsaswell asoftherelevantparametersofthe system,i.e., K and

β

=

m

/

n.Bothopen andclosedtrajectoriesare presented.The crucial point of the whole treatment, namelythe evaluationofthealgebraoftheconstants ofmotionisalsogiven here.Section4isdevotedtothespecialcaseofthemotioninthe plane.Thekeyroleplayedbytherequirementthat

β

bearational numberbecomesextremelyclearinthissomehowsimplified set-ting.Finally,insection 5, wesummarizethenewresultsthat we haveobtained,emphasizingwhatareinouropinionthemost rel-evantones.Asaclosingcomment,wewillbrieflyoutlinethemost promisingdevelopments.

2. ThePerlicksystemtypeI

TheHamiltonianforthePerlicksystemtypeIinspherical coor-dinates

(

r

,

θ,

ϕ

)

is

H±

=

β

2

(

1

+

K r2

)

p

2 r

2

+

L2

2r2

+

G

±

1

r

1

+

K r2

,

(2.1)

whereListheangularmomentumandL2canbeconsideredasan angularHamiltonianHθϕ inthevariables

(θ,

ϕ

)

,havingtheform

L2

=

Hθϕ

=

p2θ

+

p2

ϕ

sin2

θ

(2.2)

beingK,Grealconstantsand

β

=

m

/

narationalnumber.Wecan alsoconsiderp2ϕ asafreeHamiltonian inthevariables

(

ϕ

,

)

,

defined on the unit circle. The angular variables have the usual range:0

< θ <

π

and 0

<

ϕ

<

2

π

.The radial variablehas differ-ent ranges depending on K. If K

0, then 0

<

r

<

, while if K

<

0,r mustbe restrictedtothefinite interval0

<

r

<

1

/

K. Inhispaper[1]Perlick consideredonlythenegative signinfront of thesquare rootof the potential, that is

H−,in order to have boundedmotions.However, wewillshowthatforthecaseK

<

0 both Hamiltonians

H± should be taken intoaccount in orderto

have a well definedHamiltonian on a three dimensional sphere. ThemetricassociatedtothisHamiltonianis[3]

ds2

=

dr

2

β

2

(

1

+

K r2

)

+

r

2

(dθ

2

+

sin2

θ

d

ϕ

2

) .

(2.3)

AnotherpointofviewontheHamiltonians(2.1)istolookatthem not asdefinedina curvedspacebutaspositiondependent mass Hamiltoniansinaflatspace[32,33].

Since

{

L2

,

H±

}

=

0, L2 is a constant ofmotion and takes the positive valuesL2

=

2.Here,

,

·}

stands forthePoissonbrackets (PBs) thatforthefunctions f

(

r

,

θ,

φ),

g

(

r

,

θ,

ϕ

)

inspherical coor-dinatesaredefinedby

{

f

,

g

} =

f

∂r

g

pr

f

pr

g

∂r

+

f

∂θ

g

f

g

∂θ

+

f

ϕ

g

f

g

ϕ

.

(2.4)

Forthesake ofsimplicity,wealsochoose G

=

0,sowhenwe re-placeL2 byitsconstantvalue

2,wecanrewritetheinitial Hamil-tonian(2.1)asaneffectiveHamiltonian

H±r inthevariabler:

H±r

=

β

2

(

1

+

K r2

)

p

2 r

2

+

2

2r2

±

1

r

1

+

K r2

.

(2.5)

Since

{

,

H±

} =

0, isanotherconstant ofmotion:

=

z

=

const.Inthesameway,afterreplacing by

z in(2.2),wehave aneffectiveHamiltonian in

θ

:

=

p2θ

+

2z

sin2

θ

.

(2.6)

Notice that is singular attheangles

θ

=

0 and

θ

=

π

,so the trajectoriesliebetweenthesetwovalues(i.e.theNorthandSouth poles cannot be reachedunless

z

=

0). Oncefixed the valuesof

,

z, the turningpoints for

θ

are givenby the solutionsof

2

=

2

z

/

sin2

θ

.

In order totake intoaccount different signs andvaluesof K, we will make useof the

κ

-dependent cosineand sine functions definedby

Cκ

(u)

cos

κ

u

κ

>

0

1

κ

=

0

cosh

κ

u

κ

<

0

,

Sκ

(u)

1 √ κsin

κ

u

κ

>

0

u

κ

=

0

1 √

κsinh

κ

u

κ

<

0

.

(2.7)

The

κ

-tangentisdefinedby

Tκ

(u)

Sκ

(u)

Cκ

(u)

.

Somerelationsamongthese

κ

-functionsare[35,36]:

C2κ

(u)

+

κ

S2κ

(u)

=

1

,

d

duCκ

(u)

= −

κ

Sκ

(u),

d

duSκ

(u)

=

Cκ

(u) .

(2.8)

Now,letusintroducetheparameter

κ

insteadof K andmake thefollowingchangeofcanonicalvariables,

K

= −

κ

,

r

=

Sκ

(ξ ),

pr

=

Cκ

(ξ )

,

(2.9)

(3)

π

/(

2

κ

)

< ξ <

π

/

κ

)

.Thischange ofcoordinates can be inter-pretedinthefollowingway.Thevariables

(ξ,

θ,

ϕ

)

canbe consid-eredastheangularcoordinatesofthepoints ofa(pseudo)sphere in

R

4. When

κ

<

0 they parametrizethe points of one sheet of athreedimensional(3D)hyperboloid;when

κ

=

0 thecoordinates

(ξ,

θ,

ϕ

)

representthesphericalcoordinatesofthepointsof

R

3; fi-nally,for

κ

>

0 thesevariableswith0

< ξ <

π

/(

2

κ

)

parametrize thepoints ofone half (the northhemisphere) ofa deformed3D sphere,and the values

π

/(

2

κ

)

< ξ <

π

/

κ

give the points of the south hemisphere. In the case

κ

<

0, this choice of coordi-natesis notimportant sincethe northandsouthhemispheresof ahyperboloid arenot connectedandcanbe takenindependently withoutanyproblem,butinthecase

κ

>

0 thisparametrizationof thepointsofasphereisquiterelevant.

Next,we introduce a Hamiltonian H inthe variables

(ξ,

θ,

ϕ

)

definedasfollowsforanyvalueof

κ

:

H

(ξ, θ,

ϕ

)

=

β

2p

2 ξ 2

+

L2 2 1 S2 κ

(ξ )

1

Tκ

(ξ )

,

(2.10)

andthecorrespondingeffectiveHamiltonian by

=

β

2 p2ξ

2

+

2 2 1 S2 κ

(ξ )

1

Tκ

(ξ )

=

β

2p

2 ξ

2

+

Veff

(ξ ) .

(2.11)

We will see the relationof this Hamiltonian withthe Perlick Hamiltonians

H±fordifferentvaluesof

κ

.

κ

0.Inthiscase,bothvariables

ξ

andrvaryinthepositive semi-line

(

0

,

)

.Ifwechangethevariablesin(2.1)according to(2.9)itiseasytocheckthat

H

(ξ, θ,

ϕ

)

=

H

(r, θ,

ϕ

) .

(2.12)

κ

>

0. Here, there are two complementary intervals for the rangeofvariable

ξ

asmentionedabove.Ifwechangethe vari-ablesin(2.1)accordingto(2.9),weget

H

(ξ, θ,

ϕ

)

=

H

(r, θ,

ϕ

) ,

0

< ξ <

2π

κ

,

0

<

r

<

1 √

κ

,

H+

(r, θ,

ϕ

) ,

2π

κ

< ξ <

π

κ

,

1 √

κ

>

r

>

0

.

(2.13)

Therefore, inthiscasewe haveaunique HamiltonianH well defined forall the points ofthe sphere 0

< ξ <

π

/

κ

, such that onthenorthhemisphere isdescribed by

H− andonthe southby

H+.Whenthesepointsareparametrizedintermsof

(

r

,

θ,

ϕ

)

variables, eachhemisphere givesrisetothe two dis-tinctHamiltonians

H±.Thisisveryimportantsinceingeneral the trajectoriesonthe sphereare notrestricted totheupper orthelowerhemispheres,andafulldescriptionofallofthem shouldbedonebymeansofH.

Henceforth,wewillusethefollowingnotation. Forallthe val-uesof

κ

wewilltaketheHamiltonianH

(ξ,

θ,

ϕ

)

asgivenin(2.10)

ortheeffectiveHamiltonian

(ξ )

in(2.11).Therangesofthe vari-able

ξ

aretakenaccordingtothepreviousdiscussiondependingon thevaluesof

κ

,inparticularfor

κ

>

0 theintervalis0

< ξ <

π

κ.

3. Constantsofmotionfromfactorizationproperties

Inthismodel,we alreadyhavethreeindependentconstantsof motion: H

,

Hθϕ

,

with constant values given by E

,

2

,

2z. Our aimistofindadditionalconstantsofmotionbymeansofthe fac-torizationmethodinordertoshowthemaximalsuperintegrability ofthesystem. Afirstpairofconstants X± willbe obtainedfrom theeffectiveHamiltonians and,andasecondpairY± from thenexttwoeffectiveHamiltonians, and.

3.1. TheconstantsofmotionX±

The first set ofconstants X± are constructedin terms of the shift functions of and the ladder functions of which are obtainedasfollows[18].

ShiftfunctionsofHξ

TheHamiltonian(2.11)canbefactorizedas

=

B+B

+

λ

ξ

,

(3.1)

where

B±

=

1

2

i

β

+

Tκ

(ξ )

1

,

λ

ξ

= −

1 2 1

2

κ

2

.

(3.2)

The shift functions B± are complex conjugate of each other andthey satisfythefollowingPBstogetherwiththe Hamilto-nian

{

B

,

B+

} =

i

β

Sκ2

(ξ )

,

{

,

B±

} = ±

i

β

Sκ2

(ξ )

B±

.

(3.3)

ThesecondPBof(3.3)impliesthat

{

H,B±

} = ±

i

β

Hθϕ

Sκ2

(ξ )

B±

.

(3.4)

Fromthe factorizationgivenin(3.1)ortheeffectivepotential

(2.11),weconcludethattheenergyE ofthetotalHamiltonian forboundedmotions mustsatisfy the following inequali-tiesdependingon

κ

:

κ

<

0

,

|

κ

|

>

E

≥ −

1

2 1

2

+ |

κ

|

2

,

κ

=

0

,

0

>

E

≥ −

1

2

2

,

κ

>

0

,

>

E

≥ −

1

2 1

2

κ

2

.

(3.5)

In Fig. 1, it is shownthe effectivepotential Veff

(ξ )

givenby

(2.11)togetherwiththeconditionsontheenergy(3.5)for dif-ferentvaluesof

κ

.Forallthesethreecases,boundedmotions havetwoturningpointsinthevariable

ξ

givenbyE

=

Veff

(ξ )

. ForE

≥ −

|

κ

|

thetrajectorywillbeunboundedwithonlyone turningpointfor

κ

0.

LadderfunctionsofHθ

In order to find the ladder functions forthe angular Hamil-tonian ,definedby(2.6),first wemultiplyitby sin2

θ

and afterrearranging,weget

2z

=

p2θsin2

θ

sin2

θ .

(3.6)

Now,we can factorizetherighthand sideof thisequalityin termsofcomplexconjugateladderfunctions

2z

=

A+A

+

λ

θ

,

(3.7)

where

A±

= ∓

isin

θ

+

cos

θ ,

λ

θ

= −

.

(3.8)

Theseladderfunctions A± together withtheHamiltonian satisfythefollowingPBs

{

A

,

A+

} =

2i

,

{

,

A±

} = ±

2i

(4)

Fig. 1.PlotoftheeffectivepotentialVeff(ξ )forκ= −1 (left),κ=0 (center)andκ=1 (right)togetherwiththelimitingvaluesoftheenergygivenbytheinequalities(3.5)

for=0.5.Therangeofκforκ≤0 is0< ξ <∞andforκ>0 itis0< ξ <π/κ.

Therefore,thePBwithHis

{

H,A±

} = ∓

i

Hθϕ

Sκ2

(ξ )

A±

.

(3.10)

Notice that the factorization (3.7) implies the following in-equality:

2

2z

,

(3.11)

whichisreasonableinviewofthephysicalinterpretationof

(thetotal angularmomentum)and

z (thecomponentofthe angularmomentuminthez-direction).

ConstructionofX±

From the PBs (3.4) and(3.10), taking into account that

β

=

m

/

n,weobtaintheconstantsofmotionX± ofthePerlick sys-temtypeIinthefollowingform

{

H,X±

} =

0

,

X±

=

(

A±

)

m

(

B

)

n

.

(3.12)

Wecandenotethevaluesoftheseconstantsasfollows,

X±

=

qxe±x

,

0

qx

<

,

π

α

x

<

π

.

(3.13) Theabsolutevalue qx isdirectlyobtainedfromthe factoriza-tionpropertiesofA±andB±:

qx

= |

X±

| =

2

2z

m/2

E

+

1

2 1

2

κ

2

n/2

.

(3.14)

By means ofthese constants ofmotion, the relation between the variables

ξ

(or r) and

θ

along the trajectoriescan be found from (3.13) (and the relative frequencies as we will see later). However, when

=

z, thisrelation breaks, because in this case

(2.6)and(3.8)entail

=

0 and

θ

=

π

/

2.Inotherwords,if

=

z, themotiontakesplaceinthehorizontalplane z

=

0.This particu-larcasewillbeconsideredinSection4.

Inprinciple,theconstantsofmotionX±givenby(3.12)arenot polynomialinthemomentumfunctions,since A±andB±depend on

whichisasquareroot(see(2.6)).However,ifweexpandthe powers m andn in (3.12), then the real andimaginary parts of thesefunctionswillgiverisetopolynomialconstantsofmotionin themomenta[22].

3.2. TheconstantsofmotionY±

Thisnew setof constants ofmotion isderived fromtheshift functionsof andtheladderfunctionsof.Theyareobtained

inasimilarwayasthepreviousset.

ShiftfunctionsofHθ

TheangularHamiltoniandefinedby(2.6)isfactorizedas

=

p2θ

+

2z

sin2

θ

=

C

+C

+

λ

,

(3.15)

where

C±

= ∓

i pθ

+

zcot

θ ,

λ

=

2z

.

(3.16)

These factorfunctionsC± together withthe Hamiltonian satisfythefollowingPBs

{

C

,

C+

} =

2i

z

sin2

θ

,

{

,

C

±

} = ±

2i

z

sin2

θ

C

±

.

(3.17)

ThesecondPBof(3.17)impliesthat

{

Hθϕ

,

C±

} = ±

2i

sin2

θ

C

±

.

(3.18)

LadderfunctionsofHϕ

TheHamiltonian

=

p2ϕ isfactorizedas

=

D+D

,

D±

=

Hϕe

.

(3.19)

TheseladderfunctionsD± togetherwiththeHamiltonian

satisfythefollowingPBs

{

D

,

D+

} =

2i

Hϕ

,

{

Hϕ

,

D±

} = ±

2i

HϕD±

.

(3.20)

Then,thePBwiththeHamiltonianHθϕ is

{

Hθϕ

,

D±

} = ±

2i

sin2

θ

D

±

.

(3.21)

Thebuildingof Y±

The Hamiltonian Hθϕ defined by (2.6), besides , has the

constants ofmotionY±that nowcanbefoundwiththehelp oftherelations(3.18)and(3.21):

{

Hθϕ

,

Y±

} =

0

,

Y±

=

(C

±

)(

D

) .

(3.22)

The geometric meaning ofthese constants of motioncan be appreciatedbyrewritingthemintheform

Y±

= −

Lz

(L

x

±

iLy

) ,

(3.23)

where Lx, Ly and Lz are thecomponentsof theangular mo-mentuminthex, yandzdirection,respectively:

Lx

= −

sin

ϕ

cot

θ

cos

ϕ

,

Ly

=

cos

ϕ

cot

θ

sin

ϕ

,

Lz

=

.

(3.24)

Ifwedenotethevaluesoftheseconstantsofmotionby

Y±

=

qye±y

,

0

qy

<

,

π

α

y

<

π

,

(3.25)

with

qy

= |

Y±

| =

z

(

2

2z

)

1/2

,

(3.26)

(5)

Fig. 2.Plot of the trajectories ofβ=1 for the valuesE= −1 (left),E= −6 (right),κ= −1,=0.25 andz=0.1.

constant of motion fixes the relation of the angles

θ

and

ϕ

alongatrajectory.Forexample,for

y

=

0,

α

y

=

π

,therelation

(3.25)givesthefollowingexpressionfor

θ (

ϕ

)

(or

ϕ

(θ )

)

cot

θ

= −

x

z

cos

ϕ

.

(3.27)

However, asmentionedbefore,thisrelationalsobreakswhen

=

z,since inthiscase

x

=

0 andtherefore

θ

=

π

/

2.Note that,fromexpression(3.23),itisclearlyseenthatY±are poly-nomialinthemomentumvariables.

In this subsection,we have arrived atan obvious result:the HamiltonianL2 hasthesymmetriesL

zandY± orequivalently Lx,Ly andLz.Nevertheless,thebasis Lz,Y±willbethemost adequatetowritethesymmetryalgebra.

3.3.TrajectoriesofthePerlicksystemtypeI

Insummary,once fixedthe valuesofthe constants ofmotion E

,

,

z, the newconstants ofmotion X± and Y± determine the relationbetweenthecoordinatesr

θ

and

θ

ϕ

,respectively.Inthis way,onecan get allthetrajectoriesofthesystem. Now,inorder to characterize each trajectory we will use the propertiesof the ladderandshiftfunctions.

Theladder functions, B± of (3.2), andthe shiftfunctions, A± of(3.8),canbeexpressedas

B±

(ξ,

)

=

E

+

12

1 2

κ

2

1/2

e±i b(ξ,pξ)

,

A±

(θ,

)

=

2

2z

1/2e±i a(θ,pθ)

,

(3.28)

whereb

(ξ,

)

anda

(θ,

)

are realphasefunctionsthat depend alsoontheconstants ofmotion E

,

and

z.The effective Hamil-tonian givenin(2.11) dependsofthe variables

(ξ,

)

.When theenergyE satisfiestherestrictions(3.5) themotionisperiodic betweentwoturningpoints.Thevariables

(θ,

)

aredescribedby theeffectiveHamiltonian givenin(2.6).For

z

=

0,themotion ofthese variables will be periodic, the rangeof

θ

is determined by its corresponding turning points. As a consequence,the func-tionsb

(ξ,

)

anda

(θ,

)

will alsobeperiodic.Now,takinginto accounttheconstantsofmotion X± asgivenin(3.13) thephases ofA±,B±andX±arerelatedasfollows

m a(θ,pθ

)

n b(ξ,pξ

)

=

α

x

,

ξ

1

ξ

ξ

2

,

θ

1

θ

θ

2

.

(3.29)

Thisequationfixestherelationofthevariables

(ξ,

)

and

(θ,

)

alongthemotion.Therefore,ifwedifferentiate(3.29)withrespect totime,weget

ma(θ,

˙

)

nb(ξ,

˙

)

=

0

.

(3.30)

Thisimpliesthatthefrequencies

ω

ξ and

ω

θ arerelatedby

m

ω

θ

n

ω

ξ

=

0

.

(3.31)

Inthesameway,wecanwritethefunctionsC±andD± inthe form

C±

(θ,

)

=

(

2

2z

)

1/2e±i c(θ,pθ)

,

D±

(

ϕ

,

)

=

ze±i d(ϕ,pϕ)

,

(3.32)

wherec

(θ,

)

andd

(

ϕ

,

)

are realphase functions. Dueto the

constantsofmotionY±thesevariablesarerelatedby

c(θ,pθ

)

d(

ϕ

,

)

=

α

y

,

θ

1

θ

θ

2

,

π

ϕ

<

π

.

(3.33)

Asthemotionofthe

(θ,

)

variablesandthe

(

ϕ

,

)

isperiodic,

differentiating(3.33)withrespecttotime,weobtain

ω

θ

ω

ϕ

=

0

.

(3.34)

Hence,thefrequenciesoftheangularvariables

θ

and

ϕ

areequal (exceptforthecase

=

z).

Inconclusion,whentheenergyEsatisfiestherestrictions(3.5), themotionisbounded,andthefrequencies ofthethreevariables arerelatedby(3.31)and(3.34).Thisimpliesthatthebounded mo-tion isperiodic andthe trajectoriesare closed.In conclusion,we havecheckedinthiscasethattheBertrand’stheoremissatisfied.

InFig. 2andFig. 3someexamplesofthetrajectoriesfor differ-entvaluesofthe energies E andoftheparameter

β

correspond-ing to bounded and unbounded motions are shownfor the case

κ

= −

1.The trajectories ofthe initial Hamiltonian(2.1)are plot-tedinathree dimensionalspace

(

rsin

θ

cos

ϕ

,

rsin

θ

sin

ϕ

,

rcos

θ )

, where

(

r

,

ϕ

,

θ )

arethesphericalcoordinatesin

R

3.

3.4. AlgebraoftheconstantsofmotionforthePerlicksystemtypeI

As we have seen in the previous sections, we have obtained seven constants ofmotion forthe three dimensional Perlick sys-tem: H

,

Hθϕ

,

,

X±

,

Y±. Only five of these constants are

inde-pendent,forexamplewecanchoose: H

,

Hθϕ

,

, X+

,

Y+.

(6)

Fig. 3.Plot of the trajectories ofβ=2 (left),β=3 (right) for the valuesE= −6,κ= −1,=0.25 andz=0.1.

{

H,

·} =

0

,

{

Hθϕ

,

Y±

} =

0

,

{

Hθϕ

,

X±

} = ±

2i m

HθϕX±

,

{

,

Y±

} = ∓

2i

HϕY±

,

{

,

X±

} =

0

,

{

Y+

,

Y

} =

2i

Hθϕ

2

,

{

X+

,

X

} =

i m n

(H

θϕ

)

m

(

+

1 2

(

1

Hθϕ

κ

Hθϕ

))

n−1

×

(

κ

Hθϕ

+

1 Hθ3/ϕ2

)

2i m2

(H

θϕ

)

m−1

×

(H

ξ

+

1 2

(

1

Hθϕ

κ

Hθϕ

))

n

Hθϕ

,

{

X±

,

Y±

} = ∓

i m Hθϕ

+

X±Y±

,

{

X±

,

Y

} = ∓

i m Hθϕ

X±Y

.

(3.35)

Inthisway,wehavefoundthealgebraoftheconstantsofmotion foranytwocoprimeintegers m andn andanyvalueofthe con-stant

κ

. The corresponding polynomial algebras can be found as in[22].

Thecomplexconstants ofmotion X± andY±allowtogetreal constants:YR

=

Re

(

Y±

),

YI

=

Im

(

Y±

)

, XR

=

Re

(

X±

),

XI

=

Im

(

X±

)

, which will close a real algebra. Besides, for

β

=

1

,

κ

=

0, the complex constants of motion X± can be expressed in terms of the angular momentum L

=

(

Lx

,

Ly

,

Lz

)

and Runge–Lenz

A

=

(

A

x

,

A

y

,

A

z

)

vectors,

X±

=

Re

(

X±

)

±

iIm

(

X±

)

=

A

z

2

±

i

(

L

×

A

)

z

2

,

(3.36)

where

A

=

p

×

L

− ˆ

r

.

(3.37)

Here, we have used the expressions of r

,

p

,

L in spherical coor-dinates. Notice that in the context of the analysis of the Perlick system, a generalization of theRunge–Lenz vector hasbeen pro-posedinref.[4].Inthe nextsection wewillconsiderthe motion inaplanecorrespondingtospecialcase

θ

=

π

/

2.

4. SpecialcaseofthePerlicksystemtypeI

Inthissection, wewillstudythemotionofthePerlicksystem typeIinaplane,inthiswaywewanttosimplifytherelevant con-stantsofmotionandtheexpressionofthetrajectoriesdetermined bysuchconstants.

Let ustake

θ

=

π

/

2; withthis electionthe motion islimited toxy planeandtheangularmomentumalongthez-axisisequal tothetotalangularmomentum:L

=

(

0

,

0

,

Lz

) (

=

z

)

.The corre-spondingHamiltonianintheremainingr

,

ϕ

variablesis

H±

=

β

2

(

1

+

K r2

)

p

2 r

2

+

2

2r2

±

1

r

1

+

K r2

.

(4.1)

Now,we havetwotrivialconstantsofmotion:thetotal energyE and

z.FollowingthesameprocedureasinSections2,3,withthe changeofthevariables

(

r

,

pr

)

by

(ξ,

)

,wegettheHamiltonian

H(ξ, θ,

ϕ

)

=

β

2p

2 ξ

2

+

2

2 1 S2

κ

(ξ )

1

Tκ

(ξ )

,

(4.2)

withthesamerangeofthevariable

ξ

asspecifiedinSection 2.In thiscasetheeffectiveHamiltonian isgivenby

=

β

2 p2ξ

2

+

z2

2 1 S2

κ

(ξ )

1

Tκ

(ξ )

=

β

2p 2 ξ

2

+

Veff

(ξ ) .

(4.3)

It is easy to find two additional constants of motion Z± for the Hamiltonian(4.2):

{

H

,

Z±

} =

0

,

Z±

=

(

D±

)

m

(B

)

n

,

(4.4)

where

B±

=

1

2

i

β

+

z

Tκ

(ξ )

1

z

,

D±

=

Hϕe

,

(4.5)

and

=

B+B

+

λ

ξ

,

λ

ξ

= −

1 2

1

2z

κ

2z

,

(4.6)

=

p2ϕ

=

D+D

.

(4.7)

TheseconstantsofmotionZ±havecomplexvaluesdenotedby

Z±

=

qze±iϕz

,

(4.8)

where

π

ϕ

z

<

π

and0

qz

<

.Themodulusqz hasavalue determinedbytheotherconstantsofmotion E and

z,

qz

= |

Z±

| =

mz E

+

1 2

1

2z

κ

2z

n/2

(7)

Fig. 4.Plotofthetrajectoriesofβ=1 (left),β=2 (center),β=3 (right)forthevaluesE= −8.03 (minimumenergyofthepotential),E= −6,E= −1,E=4.Forβ=1, respectively.Herewehavechosenκ= −1,=0.25 andz=0.1.

ThefunctionsB±andD± canbewrittenas

B±

(ξ,

)

=

E

+

1 2

1

2z

κ

2z

1/2

e±i b(ξ,pξ)

,

D±

(

ϕ

,

)

=

ze±i d(ϕ,pϕ)

,

(4.10)

where b

(ξ,

)

and d

(

ϕ

,

)

are the corresponding real phase

functions. When the energy E satisfies the restriction (3.5), the boundedmotion of the variables

(ξ,

)

is periodic. Besides, the motionofthevariables

(

ϕ

,

)

isalsoperiodic,duetotheangular

characterof

ϕ

.Then,substitutingin(4.8)weget

m d(

ϕ

,

)

n b(ξ,pξ

)

=

ϕ

z

,

ξ

1

ξ

ξ

2

,

π

ϕ

<

π

.

(4.11)

Hence,afterdifferentiatingwithrespecttothetimeweobtainthe relationbetweenthefrequenciesofthetwovariables:

m

ω

ϕ

n

ω

ξ

=

0

.

(4.12)

Letuswritetheequationoftheconstantofmotion Z±,

(

ze

)

m

(

i

β

2

+

z

2 1 Tκ

(ξ )

1

z

2

)

n

=

q

ze±iϕz

.

(4.13)

Taking thereal andimaginary parts of thisequality, we get two equations

z

2 1 Tκ

(ξ )

1

z

2

=

qz

mz

1/n

cos 1

n

ϕ

z

+

m

n

ϕ

,

(4.14)

β

2

=

qz

mz

1/n

sin 1

n

ϕ

z

+

m n

ϕ

.

(4.15)

From (4.14), we obtain the relation between

ξ

and

ϕ

along the trajectories[4]:

cos 1

n

ϕ

z

+

m

n

ϕ

=

2z

Tκ

(ξ )

1

2E

2z

+

1

κ

4z

.

(4.16)

Relation (4.16) becomes a well knownconic section equation for thevalues

κ

=

0,

β

=

m

/

n

=

1 and

ϕ

z

=

0:

α

ξ

=

1

+

ε

cos

ϕ

,

(4.17)

where

ε

2

=

2E

2

z

+

1 (eccentricity)and

α

=

2z (semi-latusrectum)

[34].In thiscase,the problemisreducedto theKepler–Coulomb system in the Euclidean plane. If

κ

= −

1 (

κ

=

1) and

β

=

1,

ϕ

z

=

0,thenwegettheconicsectionequationinthehyperboloid (sphere)[12]:

κ

= −

1

,

(Hyperbolic)

α

tanh

ξ

=

1

+

ε

2

+

α

2cos

ϕ

,

(4.18)

κ

=

1

,

(Spherical)

α

tan

ξ

=

1

+

ε

2

α

2cos

ϕ

.

(4.19)

So, we may say that the equation (4.16) can be considered asa generalized conic section equation. Taking into account the rela-tion (4.18) for

κ

= −

1 and therelation(4.19) for

κ

=

1 different examples of trajectories

cos

ϕ

,

ξ

sin

ϕ

)

, where

(ξ,

ϕ

)

are polar coordinates, are plotted in Figs. 4–7. In these figures there are representedboundedandunboundedtrajectories,accordingtothe valuesoftheenergy E.Inthecaseofboundedmotion,duetothe above relationofthe frequencies(4.12) themotionmustbe peri-odicandthetrajectoriesareclosed.

Inthespecialcasewhere

κ

=

0 and

β

=

m

/

n

=

1,theconstants ofmotion Z± can alsobe expressedin termsof theRunge–Lenz vector

A

=

(

A

x

,

A

y

,

0

)

,

Z±

=

Re

(Z

±

)

±

iIm

(

Z±

)

=

A

x

2

i

A

y

2

.

(4.20)

Fortheother cases(

κ

= ±

1),theconstants ofmotion Z± canbe written in termsof a type ofgeneralized Runge–Lenzvector [4]. Note that, the constants H

,

,

Z± close a similar algebra as in

thegeneralcase.

5. Conclusionsandremarks

(8)

Fig. 5.Plotofthetrajectoriesofβ=1/2 (left),β=1/3 (right)forthevaluesE= −8.03 (minimumenergyofthepotential),E= −6,E= −1,E=4,correspondingtocircular, elliptic,parabolicandhyperbolicorbits,respectively.Herewehavechosenκ= −1,=0.25 andz=0.1.

Fig. 6.Plotofthetrajectoriesofβ=1 (left),β=2 (center),β=3 (right)forthevaluesE= −7.96875 (minimumenergyofthepotential),E= −3,E=0,E=1.Herewe havechosenκ=1,z=0.25.

Fig. 7.Plotofthetrajectoriesofβ=1/2 (left),β=1/3 (right)forthevaluesE= −7.96875 (minimumenergyofthepotential),E= −3,E=0,E=1.Herewehavechosen

(9)

ina perspicuousmannerfromthe graphicrepresentations ofthe (effective)potentialaswell asoftheorbits,reportedinanumber offiguresfordifferentvaluesofK and

β

.

Thefurthersteps tobeaccomplishedare clear.(i) Wehaveto seehow our construction goesover to the quantumsetting, and (ii) we have to perform a similar construction for the full Per-lickfamilyII. Workisactually inprogressinboth directions,and promisingpreliminary resultshave beenalready obtainedby the authorsandtheircollaborators.

Acknowledgements

This work was partially supported by the Spanish Minis-terio de Economía y Competitividad (MINECO) under project MTM2014-57129-C2-1-PandJuntadeCastillayLeón(VA057U16). ¸S. Kuru acknowledges Ankara University and the warm hospital-ityattheUniversidaddeValladolid,Dept.deFísica Teórica,where thisworkhasbeendone.O.Ragniscowishestothankallthe col-leaguesatDept.deFísicaTeórica,fortheirwarmhospitalityduring hisstay.

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