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
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,whereξ,
ξ
˜
∈ E
are onseutive pointsofreetiono theboundary, andv,
v
˜
the veloityvetorsatξ,
ξ
˜
after the reetion. ExpliitformulaeforB
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. Uptothesymmetry
(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)
˜
dependingonξ,
˜
v
˜
in thesamewayasL(λ)
onξ, v
,andqλ(ξ, η) =
ξ1η1
a
−
λ
+
ξ2η2
b
−
λ
.
Thebilliardtrajetorieshavethefollowinggeometrialproperties.
Eahsegmentofagivenbilliardtrajetorywithin
E
isanarofanellipsewiththeentreattheoordinateorigin[1℄. Moreover,asprovedin[13℄,alltheseellipses
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. TopologialpropertiesTheaimofthissetionistogivethetopologialharaterisationofthe
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 andC
λ
1
,C
λ
2
itsaustis,λ1
6
λ2
. Then:•
λ1
+
λ2
=
a
+
b
−
σ
2
E
;• C
λ
1
isan ellipseanditontainsE
,i.e.λ1
6
0
;•
ifasegmentofthe trajetoryontainsafousofE
,thenthenextsegmentontains 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 thedegenerate onithat oinides with the
y
-axis:b < λ2
6
a
;•
if asegmentofthe trajetorydoesnot itersetthe segment ontainingthefoi,than noneof the segmentsintersetit.
C
λ
2
is thenan ellipse withinE
:0
6
λ2
< b
.Proof. The leadingoeient of the harateristi polynomial
p(λ)
(2.3) is equaltoσ
,andtheoeientmultiplyingλ
isv
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 theseond statement.
The otherausti iseither an ellipseinsribedin thetrajetory, ahyperbola,
oradegenerateoni,whihprovestherestoftheproposition.
Wewill refertotheoni
C
λ
1
from Proposition3.1 asthe outerausti ofthegiven billiard trajetory and to
C
λ
2
as the inner austi. Proposition 3.1 showsthat theinneraustihasthepropertiesanaloguoustothepropetiesoftheunique
austiintheaseoftheelliptialbilliardwithoutpotential. Thebilliardwiththe
Hooke'spotentialhasthefamousfoalpropertyaswell.
Thephasespaeoftheelliptialbilliardis:
M
=
(ξ, v)
|
ξ
iswithinE
andv
∈
Tξ
R
with
(ξ, v1)
∼
(ξ, v2)
⇐⇒
ξ
∈ E
andv1
+
v2
∈
Tξ
E
andv1
−
v2
⊥
Tξ
E
.
Notethat
M
ontainstrajetoriesthathaveemptyintersetionwiththebound-arythese are trajetorieswith the outerausti within
E
. A trajetorywith theaustis
C
λ
1
,C
λ
2
willhaveanon-emptyintersetionwithE
ifandonlyiftheouterausti ontains
E
and theinner austiis either ellipse withinE
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
+
aξ
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
+
aξ
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)
2σ
,
p(0) = (v2ξ1
−
v1ξ2)
2
−
(bv
2
1
+
av
2
2
)
−
σ(bξ
1
2
+
aξ
2
2
) +
abσ.
Itmapseahpointof
M
B
tothepair(E, λ2)
orrespondingtothetotalenergy and theparameterof the innerausti. From Proposition 3.1, itfollowsthat theimage of
µ
istheset:n
(E, λ2)
|
0
6
λ2
6
a
andλ2
+
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
andB
are standard Fomenko atoms:A
is a level set ontaining asinglelosedorbit,while
B
denotes asingularlevelset ontaining onelosedorbitandtwoseparatrieswith homolini trajetories.
By
T
wedenotedatorus ontaining onlylosedorbits, and by8
aprodut ofgure
8
with the irlea non-standardsingular level set that ontains only losed orbits.Proof. Thelowest energylevel thatontainsbilliardtrajetoriesis
E
=
σb
2
.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
twosingular level sets. Therst oneorresponds to thedegenerate inner austi
C
a
andit ontains only oneorbit: the trajetory isplaed alongy
-axis,but now thepartilehitstheboundarywithnon-zeroveloity. Thatlevelset isrepresentedbytheFomenkoatom
A
.Theseondsingularlevelsetorrespondstotheinneraustiwiththe
parame-ter
λ2
=
a
+b
−
2
σ
E
andtheouteraustiE
. Theorbitsonthatlevelsetorrespond to theellipseswiththeentreat theorigin,whih areinsribedinE
andtouhingC
λ
2
. Amongthem,there aretwoellipsesthatdegeneratetothesegmentsontain-ingtheoriginwiththeendpointsattheintersetionpointsof
E
andC
λ
2
. Thelevelset isatorus anditisrepresentedbythenon-standardatom
T
.For
E
=
σa
2
, theinner austiorrespondingto eah non-singularlevel set isagain ahyperbola. Thenon-standardsingularlevelset, representedby thevertex
8
,orrespondstothedegenerateaustiC
b
. Theorbitsonthatlevelsetorrespondto theellipses withthe entre atthe origin,whih are insribed in
E
and ontainthe foi. One of these ellipses degenerates to the big diameter of
E
. The levelset isisomorphi to theprodut of thegure
8
withthe irle. Theorbiton the self-intersetionpartofthesetorrespondsto themotionalongthex
-axis.For
E
∈
(
σa
2
,
σ(a+b)
2
)
, there is one Liouville torus in eah of the level sets that orrespondtohyperbolasasinneraustis,andtwoLiouvilletoriiftheinneraustiisanellipse. 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 twoA
-atoms on the right hand side of the Fomenkoatom orrespond to the limit ase
when theinner austioinideswith
E
. Whentheenergyapprohesinnity, thebilliard 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. Theyorrespondtothevalueλ1
= 0
,thatisto theasewhen theouteraustiC
λ
1
oinideswiththebilliardborderE
. Alltrajetoriesmappedto thosesegmentsareellipses.
For
λ2
∈
(b, a)
, the trajetories areellipses insribed inE
and irumsribed aboutthehyperbolaC
λ
2
. Amongthem,therearetwodegenerateellipsessegmentsontainingthe originwith the endpoints at the intersetionsof
E
andC
λ
2
. Eahnon-degenerateellipseisoveredtwoorbits,orrespondingtothelokwiseandthe
ounterlokwisemotionabouttheorigin.
For
λ2
∈
(0, b)
, the trajetoriesare ellipses insribed inE
and irumsribed aboutthe ellipseC
λ
Theorrespondinglevelsetsarenon-standardatoms
T
. Saturatedneighbour-hoodsoftheseatomsin theisoenergymanifoldsareshowninFigure3.
Remark3.3. Point
λ2
=
b
,E
=
σa
2
belongstotheritialset. Thetrajetoriesareellipsesinsribedin
E
andontainingthefoi. Onlyonetrajetoryisaritialorbittheoneplaedalong
x
-axis.Theorrespondinglevelsetisanon-standardatom
8
. Itssaturatedneighbour-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 perioditrajetoriesalongtheellipseE
arestable,whileritial perioditrajetoriesalong
x
-axisareunstable.Proof. Thestatementfollowsfrom Theorem3.1,sinethetrajetoriesalong
y
-axis they orrespond toA
-atoms forλ2
=
a
, the limit trajetories alongE
toA
-atoms forλ2
= 0
, andtheritialtrajetoriesalongx
-axisare ontainedin theUsing[2℄,weangetthatonlusionfromthebifurationomplexinFigure5,
sine the points orresponding to
λ2
=
a
andλ2
= 0
are on the border of the omplex,whileinnerpointsoftheomplexorrespondtoλ2
=
b
. Remark3.4. Thestabilityofperiodiorbitsofgeneralbilliardswithpotentialisstudied in[12℄.
Aknowledgments. Theauthorisgratefultotherefereeforgreatomments
andquestions,whih ledtothesigniantimprovementofthemanusript.
TheresearhwhihhasledtothispaperwaspartiallysupportedbytheSerbian
MinistryofEduationandSiene(Projetno.174020:GeometryandTopology of
Manifolds andIntegrableDynamial Systems)andbygrantno.FL120100094from
theAustralianResearhCounil.
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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