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TOPOLOGY OF THE ELLIPTICAL BILLIARD

WITH THE HOOKE'S POTENTIAL

Milena Radnovi¢

Abstrat. UsingFomenko graphs, we presenta topologialdesription of

theelliptialbilliardwithHooke'spotential.

1. Introdution

Mathematialbilliardisasystemwhereamaterialpointmoveswithaonstant

veloity inside the billiard desk, while the impats o the boundary are ideally

elastiandtheveloityishangedaordingtothebilliardreetionlaw[15℄. The

boundaryofthebilliarddeskanbeanypieewisesmoothurve.

A remarkable lass of suh systems are elliptial billiards, due to their nie

geometrial properties. Eah trajetoryof elliptialbilliardhasaaustiaurve

touhing all segments, or their straight extensions, of the trajetory. Moreover,

the austi isaoni onfoal with theboundaryand it representsageometrial

manifestationoftheintegralofmotion[1, 15, 9℄.

Oneanonsiderbilliardmotioninapotentialeld aswell. Integrablelasses

of potential perturbations of elliptial billiards were studied in [16, 6, 7℄. The

Hooke's potential belongs to that lass and the dynamis of suh a billiard was

studiedin[13℄,wheretheLaxrepresentationofthatsystemwasonstrutedanda

generalsolutionobtainedintermsoftheta-funtions.Someremarkablegeometrial

propertiesof thesystem arealso proved in [13℄: eahtrajetoryhastwoaustis

whihareonfoalwiththeboundary.

The objetive of this work is to give topologial desription of the elliptial

billiard with the Hooke's potential. To ahieve that, we are using tools, widely

knownastheFomenkographs,whihodifythetopologialesseneoftheLiouville

foliationof integrablesystemswith twodegreesof freedom. Thedetailed aount

on Fomenkographsanbe foundin [3℄, see also [4, 5, 2℄ and referenestherein.

Fortopologial desriptionof elliptialbilliardswithoutpotentialsee [8, 10, 11℄

andforbilliardswithinonfoalparabolas[14℄.

2010MathematisSubjetClassiation: Primary37J35;Seondary70H06.

Key words and phrases: elliptial billiards, onfoal onis, integrable potential

(2)

Next,in Setion2,wereview themainstepsfrom[13℄ oftheintegration

pro-edure of the elliptial billiard within an ellipse in the presene of the Hooke's

potential. Setion 3 ontains topologial desription of that system. The main

resultisTheorem3.1,wheretheFomenkoivariantsfortheisoenergymanifoldsare

alulated. Fortheritiallosedorbits,wedisusstheirstabilityinCorollary3.1.

2. Integrability of the system

In[13℄,billiardwithinanellipsoidinthe

n

-dimensionalspaewiththeHooke's potentialwasonsidered. Inthissetion,wereviewtheintegrationproedurefrom

[13℄intheplanarase:

n

= 2

.

Supposethatin theeldwiththeHooke'spotential:

(2.1)

σ

2

(x

2

+

y

2

),

σ >

0,

apartileofunit massmoveswithinellipse:

(2.2)

E

:

x

2

a

+

y

2

b

= 1,

a > b >

0,

satisfyingthebilliardlawwhenbeingreetedo theboundary.

Let

B

: (ξ, v)

7→

( ˜

ξ,

v)

˜

bethebilliardmapping,where

ξ,

ξ

˜

∈ E

are onseutive pointsofreetiono theboundary, and

v,

v

˜

the veloityvetorsat

ξ,

ξ

˜

after the reetion. Expliitformulaefor

B

are:

˜

ξ

=

1

ν

σξ

(v, Av)ξ

+ 2(ξ, Av)v

,

˜

v

=

1

ν

σv

(v, Av)v

2σ(ξ, Av)ξ

+

µA

ξ,

˜

ν

=

p

4σ(ξ, Av)

2

+ (σ

(v, Av))

2

,

µ

=

2(˜

v, A

ξ)

˜

( ˜

ξ, A

2

ξ)

˜

,

A

=

1/a

0

0

1/b

.

Itturnsoutthatthebilliardmapping

B

hastheLaxpairrepresentation. Upto

thesymmetry

(ξ, v)

7→

(

ξ,

v)

,thebilliardmappingisequivalenttotheequation

˜

L(λ) =

M

(λ)L(λ)M

1

(λ),

where:

L(λ) =

qλ(ξ, v)

qλ(v, v)

σ

qλ(ξ, ξ) + 1

qλ(ξ, v)

,

M

(λ) =

σλ

(v, Av)λ

+ 2(ξ, Av)µ

2σ(ξ, Av)λ

σµ

+ (v, Av)µ

2(ξ, Av)λ

σλ

(v, Av)λ

,

with

L(λ)

˜

dependingon

ξ,

˜

v

˜

in thesamewayas

L(λ)

on

ξ, v

,and

qλ(ξ, η) =

ξ1η1

a

λ

+

ξ2η2

b

λ

.

Thebilliardtrajetorieshavethefollowinggeometrialproperties.

Eahsegmentofagivenbilliardtrajetorywithin

E

isanarofanellipsewith

theentreattheoordinateorigin[1℄. Moreover,asprovedin[13℄,alltheseellipses

(3)

Ifwedenotethefamilyofonisonfoalwith

E

asfollows:

C

λ

:

x

2

a

λ

+

y

2

b

λ

= 1,

then theparametersoftheaustisofagiventrajetoryaretherootsofthe

har-ateristi polynomial:

(2.3)

p(λ) = (λ

a)(λ

b) det

L(λ).

3. Topologialproperties

Theaimofthissetionistogivethetopologialharaterisationofthe

isoen-ergymanifolds.

Inthenextproposition,wesummarisesomegeometrialpropertiesofthe

sys-tem.

Proposition3.1. Consideragivenbilliardtrajetorywithintheellipse

E

(2.2)

with the Hooke's potential (2.1). Let

E

be the total energy orresponding to that trajetory and

C

λ

1

,

C

λ

2

itsaustis,

λ1

6

λ2

. Then:

λ1

+

λ2

=

a

+

b

σ

2

E

;

• C

λ

1

isan ellipseanditontains

E

,i.e.

λ1

6

0

;

ifasegmentofthe trajetoryontainsafousof

E

,thenthenextsegment

ontains theother fous.

C

λ

2

isthenthe degenerate oni:

λ2

=

b

;

if a segment of the trajetory itersets the segment ontaining the foi,

than eah segment intersets it as well.

C

λ

2

is then a hyperbola or the

degenerate onithat oinides with the

y

-axis:

b < λ2

6

a

;

if asegmentofthe trajetorydoesnot itersetthe segment ontainingthe

foi,than noneof the segmentsintersetit.

C

λ

2

is thenan ellipse within

E

:

0

6

λ2

< b

.

Proof. The leadingoeient of the harateristi polynomial

p(λ)

(2.3) is equalto

σ

,andtheoeientmultiplying

λ

is

v

2

+

σξ

2

σ(a

+

b) = 2E

σ(a

+

b)

,

from wheretherststatmentfollows.

Oneoftheaustiswillbetheellipseontainingtheextensionsofthebilliard

segments. Sine the billiard impats are o

E

, that austi need to satisfy the

seond statement.

The otherausti iseither an ellipseinsribedin thetrajetory, ahyperbola,

oradegenerateoni,whihprovestherestoftheproposition.

Wewill refertotheoni

C

λ

1

from Proposition3.1 asthe outerausti ofthe

given billiard trajetory and to

C

λ

2

as the inner austi. Proposition 3.1 shows

that theinneraustihasthepropertiesanaloguoustothepropetiesoftheunique

austiintheaseoftheelliptialbilliardwithoutpotential. Thebilliardwiththe

Hooke'spotentialhasthefamousfoalpropertyaswell.

Thephasespaeoftheelliptialbilliardis:

M

=

(ξ, v)

|

ξ

iswithin

E

and

v

R

(4)

with

(ξ, v1)

(ξ, v2)

⇐⇒

ξ

∈ E

and

v1

+

v2

E

and

v1

v2

E

.

Notethat

M

ontainstrajetoriesthathaveemptyintersetionwiththe

bound-arythese are trajetorieswith the outerausti within

E

. A trajetorywith the

austis

C

λ

1

,

C

λ

2

willhaveanon-emptyintersetionwith

E

ifandonlyiftheouter

austi ontains

E

and theinner austiis either ellipse within

E

orahyperbola,

whih is equivalent to

λ1λ2

6

0

, i. e. the onstant term of the harateristi polynomial(2.3)needtobenon-positive:

(v2ξ1

v1ξ2)

2

(bv

1

2

+

av

2

2

)

σ(bξ

1

2

+

2

2

) +

abσ

6

0.

So,wewill onsiderthefollowingsubsetof

M

:

M

B

=

{

(ξ, v)

∈ M |

(v2ξ1

v1ξ2)

2

(bv

1

2

+

av

2

2

)

σ(bξ

1

2

+

2

2

) +

abσ

6

0

}

.

Denote by

µ

bethefollowingmap:

µ

:

M

B

R

2

,

(ξ, v)

7→

(E, λ2),

with

E(ξ, v) =

v

2

2

+

σξ

2

2

,

λ2(ξ, v) =

σ(a

+

b)

v

2

σξ

2

+

p

(σ(a

+

b)

v

2

σξ

2

)

2

4σp(0)

,

p(0) = (v2ξ1

v1ξ2)

2

(bv

2

1

+

av

2

2

)

σ(bξ

1

2

+

2

2

) +

abσ.

Itmapseahpointof

M

B

tothepair

(E, λ2)

orrespondingtothetotalenergy and theparameterof the innerausti. From Proposition 3.1, itfollowsthat the

image of

µ

istheset:

n

(E, λ2)

|

0

6

λ2

6

a

and

λ2

+

2

σ

E

>

a

+

b

o

,

whihisshowninFigure1.

Theorem 3.1. The isoenergy manifolds for the billiard within the ellipse

E

with the Hooke'spotential (2.1)areshownin Figure 2.

There

A

and

B

are standard Fomenko atoms:

A

is a level set ontaining a

singlelosedorbit,while

B

denotes asingularlevelset ontaining onelosedorbit

andtwoseparatrieswith homolini trajetories.

By

T

wedenotedatorus ontaining onlylosedorbits, and by

8

aprodut of

gure

8

with the irlea non-standardsingular level set that ontains only losed orbits.

Proof. Thelowest energylevel thatontainsbilliardtrajetoriesis

E

=

σb

2

.

(5)

λ2

0

b

a

E

σb

2

σa

2

σ(a+b)

2

Figure 1. Bifurationdiagramfortheelliptialbilliardwiththe

Hooke'spotential.

λ2

= 0

λ2

=

b

λ2

=

a

A

E

=

σb

2

A T

σb

2

< E <

σa

2

A

8

E

=

σa

2

A B

T T

σa

2

< E <

σ(a+b)

2

A B

A A

r

= 0

ε

= 1

r

=

0

ε

=

1

r

=

0

ε

=

1

E

>

σ(a+b)

2

n

=

1

Figure2. TheFomenkographsfortheelliptialbilliardwiththe

Hooke'spotential.

For

E

(

σb

2

,

σa

(6)

twosingular level sets. Therst oneorresponds to thedegenerate inner austi

C

a

andit ontains only oneorbit: the trajetory isplaed along

y

-axis,but now thepartilehitstheboundarywithnon-zeroveloity. Thatlevelset isrepresented

bytheFomenkoatom

A

.

Theseondsingularlevelsetorrespondstotheinneraustiwiththe

parame-ter

λ2

=

a

+b

2

σ

E

andtheouterausti

E

. Theorbitsonthatlevelsetorrespond to theellipseswiththeentreat theorigin,whih areinsribedin

E

andtouhing

C

λ

2

. Amongthem,there aretwoellipsesthatdegeneratetothesegments

ontain-ingtheoriginwiththeendpointsattheintersetionpointsof

E

and

C

λ

2

. Thelevel

set isatorus anditisrepresentedbythenon-standardatom

T

.

For

E

=

σa

2

, theinner austiorrespondingto eah non-singularlevel set is

again ahyperbola. Thenon-standardsingularlevelset, representedby thevertex

8

,orrespondstothedegenerateausti

C

b

. Theorbitsonthatlevelsetorrespond

to theellipses withthe entre atthe origin,whih are insribed in

E

and ontain

the foi. One of these ellipses degenerates to the big diameter of

E

. The level

set isisomorphi to theprodut of thegure

8

withthe irle. Theorbiton the self-intersetionpartofthesetorrespondsto themotionalongthe

x

-axis.

For

E

(

σa

2

,

σ(a+b)

2

)

, there is one Liouville torus in eah of the level sets that orrespondtohyperbolasasinneraustis,andtwoLiouvilletoriiftheinner

austiisanellipse. Thenon-standardsingularlevelsetontainslosedorbitsthe

trajetoriesareellipses insribedin

E

and irumsribed aboutthe inner austi.

That set is omposed of twotorieah one orresponds to one winding diretion

abouttheorigin.

When

E

>

σ(a+b)

2

there are nonon-standardsingular levelsets. The two

A

-atoms on the right hand side of the Fomenkoatom orrespond to the limit ase

when theinner austioinideswith

E

. Whentheenergyapprohesinnity, the

billiard segmentsbeome straight, sothe numerialtopologial invariantswill be

thesameasintheaseofthebilliardwithoutpotential.

Remark3.1. Thebifurationsetofthesystemistheunionofthreehalf-lines:

λ2

=

a, E

>

σb

2

,

λ2

=

b, E

>

σa

2

,

λ2

= 0, E

>

σ(a

+

b)

2

,

see Figure1.

Remark 3.2. Thesegments

λ2

=

a

+

b

2

σ

E

(0, b)

(b, a)

doesnotbelong to theiritialset. Theyorrespondtothevalue

λ1

= 0

,thatisto theasewhen theouterausti

C

λ

1

oinideswiththebilliardborder

E

. Alltrajetoriesmapped

to thosesegmentsareellipses.

For

λ2

(b, a)

, the trajetories areellipses insribed in

E

and irumsribed aboutthehyperbola

C

λ

2

. Amongthem,therearetwodegenerateellipsessegments

ontainingthe originwith the endpoints at the intersetionsof

E

and

C

λ

2

. Eah

non-degenerateellipseisoveredtwoorbits,orrespondingtothelokwiseandthe

ounterlokwisemotionabouttheorigin.

For

λ2

(0, b)

, the trajetoriesare ellipses insribed in

E

and irumsribed aboutthe ellipse

C

λ

(7)

Theorrespondinglevelsetsarenon-standardatoms

T

. Saturated

neighbour-hoodsoftheseatomsin theisoenergymanifoldsareshowninFigure3.

Remark3.3. Point

λ2

=

b

,

E

=

σa

2

belongstotheritialset. Thetrajetories

areellipsesinsribedin

E

andontainingthefoi. Onlyonetrajetoryisaritial

orbittheoneplaedalong

x

-axis.

Theorrespondinglevelsetisanon-standardatom

8

. Itssaturated

neighbour-hoodisshowninFigure 4.

×

S

1

Figure 3.

Neighbourhoodof

T

.

×

S

1

Figure 4.

Neighbourhood of

8

.

Along with the bifuration diagram, it is useful to onstrut the bifuration

omplex, thatis thetopologial spae,whose pointsare onnetedomponentsof

the level sets in the Liouville foliation,with the naturalfator-topology[2℄. The

bifuration omplexfor the elliptial billiard with the Hooke's potentialis aell

omplexomposed ofthree ellswithaommonhalf-line,asshowninFigure 5.

Figure 5. Bifuratonomplex.

Corollary3.1. Critialperiodibilliardtrajetoriesalong

y

-axisandthelimit perioditrajetoriesalongtheellipse

E

arestable,whileritial perioditrajetories

along

x

-axisareunstable.

Proof. Thestatementfollowsfrom Theorem3.1,sinethetrajetoriesalong

y

-axis they orrespond to

A

-atoms for

λ2

=

a

, the limit trajetories along

E

to

A

-atoms for

λ2

= 0

, andtheritialtrajetoriesalong

x

-axisare ontainedin the

(8)

Using[2℄,weangetthatonlusionfromthebifurationomplexinFigure5,

sine the points orresponding to

λ2

=

a

and

λ2

= 0

are on the border of the omplex,whileinnerpointsoftheomplexorrespondto

λ2

=

b

.

Remark3.4. Thestabilityofperiodiorbitsofgeneralbilliardswithpotential

isstudied in[12℄.

Aknowledgments. Theauthorisgratefultotherefereeforgreatomments

andquestions,whih ledtothesigniantimprovementofthemanusript.

TheresearhwhihhasledtothispaperwaspartiallysupportedbytheSerbian

MinistryofEduationandSiene(Projetno.174020:GeometryandTopology of

Manifolds andIntegrableDynamial Systems)andbygrantno.FL120100094from

theAustralianResearhCounil.

Referenes

1. V.I.Arnold,MathematialMethodsofClassialMehanis,SpringerVerlag,NewYork,1978.

2. A.V. Bolsinov, A.V. Borisov, I.S. Mamaev, Topology and stability of integrable systems,

RussianMath.Surveys65(2010),259317.

3. A.V.Bolsinov,A.T.Fomenko,IntegrableHamiltonianSystems: Geometry,Topology,

Clas-siation,ChapmanandHall/CRC,BoaRoton,Florida,2004.

4. A.V.Bolsinov,S.V.Matveev,A.T.Fomenko,Topologiallassiation ofintegrable

Hamil-toniansystemswith twodegreesof freedom.Listofsystemswithsmall omplexity,Russian

Math.Surveys45(1990),5994.

5. A.V.Bolsinov,A.A.Oshemkov, Singularitiesof integrableHamiltonian systems,in:

Topo-logialMethodsintheTheoryofIntegrableSystems,CambridgeSientiPubl.2006,167.

6. V.Dragovi¢,OnintegrablepotentialperturbationsoftheJaobiproblemforthegeodesison

theellipsoid,J.Phys.A:Math.Gen.29(1996),L317L321.

7. , The Appell hypergeometri funtions and lassial separable mehanial systems,

J.Phys.A:Math.Gen.35(2002),22132221.

8. V.Dragovi¢,M.Radnovi¢,BifurationsofLiouvilletoriinelliptialbilliards,Regul.Chaoti

Dyn.14(2009),479494.

9. ,PoneletPorismsandBeyond,SpringerBirkhauser,Basel,2011.

10. ,Minkowskiplane,onfoalonis,andbilliards,Publ.Inst.Math.,Nouv.Sér.94(108)

(2013),1730.

11. , Topologial invariants for elliptial billiards and geodesis on ellipsoids in the

Minkowskispae,Fundam.Si.Appl.20(2015).

12. H.R.Dullin,Linearstabilityinbilliardswithpotential,Nonlinearity11(1998),151173.

13. Y. Fedorov, An ellipsoidal billiard with quadratipotential,Funt. Anal. Appl.35(2001),

199208.

14. V.V.Fokiheva,Classiationofbilliardmotionsindomainsboundedbyonfoalparabolas,

Sbornik: mathematis205(2014),12011221.

15. V.Kozlov,D.Treshhëv,Billiards,Amer.Math.So.,ProvideneRI,1991.

16. V.V.Kozlov,SomeintegrablegeneralizationsoftheJaobiproblemongeodesisonan

(9)

TOPOLOGIJA ELIPTIQKOG BILIJARA SA HUKOVIM

POTENCIJALOM

Rezime.

Koristei Fomenkove grafove, dajemo topoloxki opis

elipti-qkog bilijara sa Hukovim potencijalom.

MathematialInstituteSANU (Reeived03.07.2015)

KnezaMihaila36,Belgrade (Revised27.08.2015)

Serbia

ShoolofMathematisandStatistis

UniversityofSydney,NSW

Australia

milenami.sanu.a.rs

(10)

Imagem

Figure 2. The F omenko graphs for the ellipti
al billiard with the
Figure 5. Bifur
aton 
omplex.

Referências

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