Vetor Constants of Motion for Time-Dependent
Kepler and Isotropi Harmoni Osillator Potentials
O.M. Ritter
;
Departamento deFsia,
UniversidadeFederalde SantaCatarina,Trindade
88040-900 FlorianopolisSC,Brasil
F.C. Santos y
, and A. C. Tort z
InstitutodeFsia,Universidade FederaldoRiodeJaneiro
CidadeUniversitaria,CaixaPostal68528
21945-970Riode Janeiro RJ,Brasil
Reeivedon8June,2000. Revisedversionreeivedon21Marh,2001.
Wedisussamethodfor obtainingvetoronstantsofmotionfor time-dependententral elds.
I Introdution
The searh for rst integralshas assumed an
inreas-ing importane in thedetermination of the
integrabil-ityofadynamialsystem. Thenotionof integrability
is related to the existene of rst integralsof motion.
Several methods ofnding rst integralsare available
in theliterature. Someofthem arebasedonthe
sym-metries of the system [1℄, [2℄. Another possibility
re-lies on a diret searh for invariants with a speied
form(usually,apolynomialformontheveloities)[3℄,
[4℄. Still other methods to determinethe integrability
ofasystemareavailableas,forinstane, thePainleve
Analysis [5℄. One of the most aesthetially appealing
and important problem in physis is the entral eld
problem [6℄. Energy and angular momentum
assoi-ated with this type of eld are well known onserved
quantities. However,othervetorandtensoronserved
quantities have been assoiated with some partiular
entral elds. TheLaplae-Runge-Lenz vetor[7℄ is a
vetorrstintegralof motionfor theKepler problem;
the Fradkin tensor [8℄ is onserved forthe aseof the
harmoniosillatorandforanyentraleld itis
possi-bletondavetorrstintegralofmotionaswasshown
in [9℄. Inthegeneralasetheseadditionalintegralsof
motionturnouttobeompliatedfuntionsofthe
po-sitionrandlinearmomentumpofthepartileprobing
theentraleld. Whenorbitsarelosed and
degener-ated with respet to the mehanial energy, however,
weshouldexpetthese additionalonstantsof motion
tobesimplefuntionsofrandp. Inthisartilewewish
toexploit furtherthislineofreasoningbydetermining
theexisteneofsuhadditionalvetorrstintegralsof
motionforsomespeiallassoftime-dependentKepler
and isotropiharmoni osillator problems. In
parti-ular, wewill show that forthetime-dependentKepler
problem the existene of a vetor onstant of motion
oupled to a simple transformation of variables turns
theproblemeasilyintegrable.
II Construting vetor
on-stants of motion
Theforef(r;t)atingonapartilemovinginaentral
but otherwise arbitrary and possible time-dependent
eld offoreg(r;t)anbewrittenas
f(r;t)=g(r;t)r; (1)
wherer=r( t)isthepositionvetorwithrespettothe
enteroffore,risitsmagnitude,andtisthetime.
As-sumingaonstantvetorj oftheform
j(p;r;t)=A( p;r;t)p+B( p;r;t)r; (2)
e-mail:fs1omrsia.ufs.br
y
e-mail:ladelfif.ufrj.br
wherep=p(t)=mr(t)_ isthelinearmomentum,mis
the redued massand A, B are arbitrarysalar
fun-tionsof p;r andt,it followsthat thefuntions Aand
B mustsatisfy
Ag+ dB
dt
=0; (3)
dA
dt +
B
m
=0: (4)
EliminatingB between(3)and(4)weobtain
m d
2
A
dt 2
gA=0: (5)
It followsfrom(4)that j anbewritten intheform
j=Ap m dA
dt
r: (6)
Iftheeldg(r;t)isknownanysolutionof(5)willyield
a vetor onstant of motion of the form given by (6).
Equation(5),however,isadierentialequationwhose
solutionmayturnouttobeahardtasktoaomplish.
Nevertheless,weanmakeprogressifinsteadoftrying
totakleitdiretlywemakeplausibleguesses
onern-ing A thereby linking j to spei forms of the eld
g(r;t).
III Time-dependent elds
Let us onsider time-dependent entral fore elds for
whihweanbuildmoregeneralvetorrstintegralsof
motion. Aswiththetime-independentasethereareof
ourseseveralpossibilitieswhenitomestothehoie
ofafuntionAforatime-dependententraleld. Here
isone
A=(t)rp+ (t)rr: (7)
Using(7)in (5)andassumingtheadditionalondition
+m d
dt
=0; (8)
wearriveat
g(r;t)= m
d
2
dt 2
+ C
r 3
: (9)
Thevetorrst integralofmotionassoiatedwith (9)
is
j=m 2
L d
dt
r
mCr
r
; (10)
where we have made use of the fat that the angular
momentum is onstant for any arbitrary entral eld
whetheritistime-independentornot. Ifin(10)weset
=1andC6=0,wean reoverthetime-independent
Kepler problem. If we set (m=)d 2
=dt 2
= k(t);
that is, if is an arbitrary funtion of the time, and
alsoC=0,wehavethetime-dependentisotropi
har-moni osillator eld, F( r) = k(t)r. In this ase
j is equal to the rst term on the R.H.S. of (10). If
(m=)d 2
=dt 2
= k and C = 0, we have the
time-independentisotropiharmoniosillatoreld. Finally,
if(m=)d 2
=dt 2
=0;6=0foranyinstantoftimetand
C6=0;wehaveatime-dependentKeplereld. Inthis
ase
g(r;t)= C
(
1 t+
0 )r
3
; (11)
andtheassoiatedonstantvetorisatime-dependent
generalizationoftheLaplae-Runge-Lenzvetor
j=m(
1 t+
0 )
2
L
dr=dt
1 t+
0
1 r
(
1 t+
0 )
2
mCr
r
: (12)
d
As another example let us onsider the
time-dependentisotropiharmoniosillator andshowhow
it is possible to generalize the Fradkin tensor for this
ase. Letthefuntion A(r;p;t)bewrittenas
A=(t)^ur+ (t)u^p: (13)
Using(5)and assumingtheadditionalondition
2 d
dt +m
d 2
dt 2
=0; (14)
weobtain
m d
3
3 4g
d
2 dg
=0: (15)
Weansolve(15)thoroughlyifg(r;t) isafuntion of
thetimetonly. Inthisase,asbefore,weendupwith
thetime-dependentisotropiharmoniosillator.From
(13)and(14)weanderivethevetorjassoiatedwith
(13)whih,intermsofomponentsreads
j
i =F
ij u
j
; (16)
where F
ij
is a generalizedFradkin tensor and anbe
F
ij =
m
2 d
dt
p
i x
j + p
i p
j +
m 2
2 d
2
dt 2
mg
x
i x
j m
2 d
dt x
i p
j
: (17)
d
It is not hard to see that the trae of this
gener-alized Fradkin tensor F(r;p;t)beomesthe energyof
thepartilewheng(r;t)isaonstanteld.
IV Obtaining expliit solutions:
An alternative way
Nowwewill showhowtotakeadvantageofthevetor
onstantjtoobtainthesolutionfortheKeplerandthe
isotropiharmoniosillatorpotentials. Butrstlywe
must establish some very general relationships. First
notiethat (6)anbereasted intotheform
j=mA(r;p;t) 2
d
dt
r
A( r;p;t)
: (18)
Integrating(18)wereadilyobtain
r( t)
A(r;p;t)
r( 0)
A(r( 0);p(0);0) =
j
m Z
t
0
d
A( r();p();) 2
: (19)
d
Letusnowshowhowweanobtaintheorbitsinthe
ase of the time-dependent Kepler problem. In what
follows,wewill bedealingwithaspeiallassof
time-dependent Kepler problems, namely, those satisfying
(9). In polar oordinatestheangular momentum
on-servationlawiswrittenintheform
l=mr 2
d
dt
(20)
andthisallowstorewrite(18)as
j=l
d
d
A
r
^
r+ A
r ^
; (21)
where wehaveintrodued thepolar unitary vetors ^
r
and ^
. Theomponentofthevetorjin thediretion
of ^
isgivenby
j ^
=l A
r
: (22)
In setion 3 we determined a generalized
Laplae-Runge-Lenzvetorforthetime-dependentKepler
prob-lem. ThesalarfuntionA( r;p;t)assoiatedwiththis
vetorwasfoundtobe
A=(t) rp m d
r 2
: (23)
ConsideringAasafuntionofweanwrite
A
r = l
d
d
r
: (24)
Equations(22)and(24)leadto
d
d
r
= j
^
l 2
= j
l 2
sin( ) ; (25)
whereistheanglebetweenjandtheOX axis. Upon
integrating(25)wend
r =
0
r
0 +
j
l 2
[os(
0
) os( )℄ : (26)
Thisistheorbitequationwewerelookingfor.
It ispossibleto transform thetime-dependent
Ke-pler problem into the time-independent one. Inorder
toseethiswedeneanewpositionvetorr 0
aording
to
r 0
= r
; (27)
andredeneourtimeparameteraordingto
dt 0
= dt
2
We an reast the equation of motion for the
time-dependentKeplerproblem,namely
m d
2
r
dt 2
=
m
d
2
dt 2
+ C
r 3
r (29)
into a simpler form. Making use of the above
trans-formationstheequationofmotion(29)an bewritten
as
m d
2
r
dt 02
0
= C
(r 0
) 3
r 0
: (30)
Equation (30) orresponds to the usual
time-independentKepler problem. Equations(27)and(28)
show that the open solutions of (30) are transformed
into the open solutions of (29) with the same
angu-larsizeandthatlosedsolutionsof(30)areassoiated
with spiraling solutions of (29). To onludeonsider
thetotalmehanialenergyassoiatedwith(30)
E= p
02
2m +
C
r 0
=onst: (31)
Sinep 0
andparerelatedby
p 0
=(t)p m _
(t)r (32)
andr 0
andrby(27)weeasily obtain
E= 2
p 2
2m 2
d
dt rp+
d
dt
2
r 2
2 +C
r
(33)
whih is a onserved quantity and an be interpreted
asa generalizationof theenergyof the partileunder
theationofatime-dependentKeplereld.
V Conlusions
In this paper we have outlined a simple and
ee-tive method for treating problems related with
time-dependent and time independent entral fore elds.
andthe isotropiharmoni osillator elds. The
diÆ-ulty in nding vetoronstantsof motion forentral
elds stems from the fat that, in general, orbits for
thistypeofproblem arenotlosed,thereforeanynew
ways to attak those problems are welome. In our
methodthisdiÆultyistransferred,sotospeak,tothe
obtentionfor eah possibleentral eld, whih anbe
time-dependent ornot, of aertain salar funtion of
theposition,linear momentum and time. Fora given
entraleldthis salarfuntion isasolutionof(5). In
thegeneralase,theobtentionofthesalarfuntion is
adiÆult task. Judiious guesses, however, failitate
thesearhfor solutionof (5)and thisiswhat wehave
donehere. Notiealsothatthemethodaplliestolosed
orbits
0
<1aswell asopenorbits
0
>1. Thease
0
=1,theparaboliorbitisalsoontemplated.
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