Restricted four and five body problemsin the Solar System
Frederic Gabern andAngel Jorba
CDS Seminar.October 8, 2003.
Frederic Gabern
Applied Mathematics and Analysis Control and Dynamical Systems
University of Barcelona California Institute of Technology
1
Contents
• Introduction.
– The Trojan asteroids.
– Review of the existing results.
– Motivation for the present work.
• Periodic models.
– The Bicircular Coherent Problem.
– The Elliptic Restricted Three Body Problem.
• Quasi-periodic models.
– The Bianular Problem.
– The Tricircular Coherent Problem.
• Conclusions.
2
The Trojan asteroids (I)
Sun
Jupiter
60
60
L5: Trojan asteroids
L4: Greek asteroids
L4: L5:
Achilles Patroclus
Hector Aeneas
Nestor Memnon
Agamemnon Paris
3
The Trojan asteroids (II)
4
The Trojan asteroids (III)
5
Previous Work
The Restricted Three Body Problem
• Analytic Tools.
• Semi-analytic tools.
Theoretical Results
mThe Outer Solar System
• Frequency analysis.
• Lyapunov exponents.
Numerical Results
6
The Restricted Three Body Problem
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Thedynamicsaround the triangular points of theSun-Jupitersystem has
been studied using analytical and semi-analytical tools (transformation of
the Hamiltonian to anormal form, computation of some approximatefirst
integrals) by several authors:Giorgilli, Delshams, Fontich, Galgani &
Simo, 1989;Simo, 1989;Celletti & Giorgilli , 1991;Jorba & Simo, 1994
(in the elliptic case);Giorgilli & Skokos, 1997;Skokos & Dokoumetzidis,
2000.
7
Realistic Model: OSS
The dynamics of some jovianTrojan asteroidshas been studied in several
papers using theOuter Solar System: Bien & Schubart, 1987;Milani,
1993;Levison, Shoemaker & Shoemaker, 1997;Pilat-Lohinger, Dvorak
& Burger, 1999;Tsiganis, Dvorak & Pilat-Lohinger, 2000.
8
Some New Models
We want to build and study models more sophisticated than the RTBP.
These models will try tosimulatein a better way the relativeSun-Jupiter
motion and they will be written as aRTBP perturbationin order the
semi-analytical tools can be applied.
In this work, we focus on the following models:
• TheBicircular Coherent Problem(BCCP).
• TheElliptic Restricted Three Body Problem(ERTBP).
• TheBianular Problem(BAP).
• TheTricircular Coherent Problem(TCCP).
9
The BCCP Model (I)
• We look for aperiodicsolution of the planar Sun-Jupiter-Saturn
Three Body Problem.
• Theperiodis chosen to be the relative period ofSaturnin the
Sun-Jupiter RTBP system:Tsat.
• The osculating eccentricity ofJupiteris small (about 50 times smaller
than the real one) but thesemi-major axisand theperiodare quite
well adjusted to the actual ones.
10
The BCCP Model (II)
It is possible to write the equations of a massless particle that moves underthe attraction of the three primaries. The correspondingHamiltonianis:
HBCCP =1
2α1(θ)(p
2x + p2
y + p2z) + α2(θ)(xpx + ypy + zpz)
+α3(θ)(ypx − xpy) + α4(θ)x + α5(θ)y − α6(θ)
[1− µ
qS+
µ
qJ+
msat
qsat
]where
q2S = (x − µ)2 + y2 + z2,
q2J = (x − µ + 1)2 + y2 + z2,
q2sat = (x − α7(θ))
2 + (y − α8(θ))2 + z2,
andθ = ωsatt + θ0. The auxiliaryαi(θ) functions are2π-periodic and are
obtained by means of Fourier analysis.
Note: The construction and study of the BCCP model can be found inGabern &
Jorba, DCDS–B 1:2, 2001.
11
The ERTBP
Theelliptic RTBPis a classical model for studying the dynamics of asmall particle in the Sun-Jupiter system. We rewrite it in such a way thatthephysical timeis the independent variable.
HERTBP =1
2α1(θ)(p
2x + p2
y + p2z) + α2(θ)(xpx + ypy + zpz)
+α3(θ)(ypx − xpy)− α4(θ)
[1− µ
qS+
µ
qJ
]where
q2S = (x − µ)2 + y2 + z2,
q2J = (x − µ + 1)2 + y2 + z2,
θ = t + θ0, and
αj(θ) =∑k≥0
αjk cos(kθ), j = 1, 3, 4 ; α2(θ) =∑k≥1
α2k sin(kθ),
are2π-periodicfunctions that are obtained from the elliptic orbit.
12
Local Study aroundL5
In theBCCPandERTBPsystems, theL5 point is replaced by aperiodic
orbit.
0.8645
0.865
0.8655
0.866
0.8665
0.867
0.8675
-0.501 -0.5005 -0.5 -0.4995 -0.499 -0.4985 -0.498 -0.4975 -0.497
(BCCP case: x-y projection)
The linear dynamics around these orbits is totallyelliptic.
13
Expansion of the Hamiltonian
We compose three linear changes of variables:
1. A PeriodicTranslation.
2. A Symplectic FloquetTransformation.
3. A Complexification.
to write the second degree of the Hamiltonians as:
H2(q, p) = iω1q1p1 + iω2q2p2 + iω3q3p3.
The frequencies are
BCCP ERTBP
ω1 -0.08047340341466 -0.08080364304385
ω2 0.99668687782956 0.99675885471089
ω3 1.00006744139040 1.0
Finally, we expand the Hamiltonians inFourier-Taylorseries.
14
Truncated Normal Forms
Using theLie series methodimplemented as inJorba 99, we transform
the expanded Hamiltonians to truncated normal forms up to high degree:
HBCCP = ωsatIθ + ω1I1 + ω2I2 + ω3I3
+
[N/2]∑n=2
H(n)(I1, I2, I3) +∑k≥N
Hk(q, p, θ)
HERTBP = Iθ + ω1I1 + ω2I2 + I3
+
[N/2]∑n=2
H(n)(I1, I2, I3, ϕ3 − θ) +∑k≥N
Hk(q, p, θ)
whereI1, I2 andI3 are the actionsIj = qj · pj .
Iθ andIθ are the “fictitious” momenta corresponding to the variablesθ = t + θ0
andθ = ωsatt + θ0.
15
Local Non-Linear Dynamics (I)
• BCCP: The phase space around the periodic orbit is completely
foliated by 1, 2 and 3-parametric families ofinvariant tori. In
particular, in the initial BCCP coordinates, they give rise to2, 3 and
4-dimensional tori(the perturbation adds an extra frequency).
• ERTBP: (Jorba & Simo, 94) The phase space
(J3, ψ3) = (Iθ + I3, ϕ3 − θ)
corresponds essentially to a perturbed pendulum depending on the
parametersI1 andI2. In particular, near the periodic orbit replacing
L5 we havenormally hyperbolic invariant toriof dimensions2 and3.
16
Local Non-Linear Dynamics (II): ERTBP
0.84
0.845
0.85
0.855
0.86
0.865
0.87
0.875
0.88
0.885
0.89
-0.54 -0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47 -0.46-0.58
-0.56
-0.54
-0.52
-0.5
-0.48
-0.46
-0.44
-0.42
-0.91 -0.9 -0.89 -0.88 -0.87 -0.86 -0.85 -0.84 -0.83 -0.82
0.858
0.86
0.862
0.864
0.866
0.868
0.87
0.872
0.874
-0.508 -0.506 -0.504 -0.502 -0.5 -0.498 -0.496 -0.494 -0.492 -0.49-0.55
-0.54
-0.53
-0.52
-0.51
-0.5
-0.49
-0.48
-0.47
-0.46
-0.45
-0.89 -0.885 -0.88 -0.875 -0.87 -0.865 -0.86 -0.855 -0.85 -0.845 -0.84
0.845
0.85
0.855
0.86
0.865
0.87
0.875
0.88
0.885
-0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47-0.58
-0.56
-0.54
-0.52
-0.5
-0.48
-0.46
-0.44
-0.42
-0.91 -0.9 -0.89 -0.88 -0.87 -0.86 -0.85 -0.84 -0.83 -0.82
17
Local Non-Linear Dynamics (III): BCCP
0.86
0.861
0.862
0.863
0.864
0.865
0.866
0.867
-0.503 -0.502 -0.501 -0.5 -0.499 -0.498 -0.497 -0.496 -0.495-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
-0.55 -0.54 -0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47 -0.46 -0.45 -0.440.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
0.91
-0.56 -0.54 -0.52 -0.5 -0.48 -0.46 -0.44
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
-0.54 -0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47 -0.460.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
-0.55 -0.54 -0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47 -0.46 -0.45
18
Approximate First Integrals
Given the expanded Hamiltonians
HM(q, p, θ, pθ) = ωMpθ + HM2 (q, p) +∑n≥3
HMn (q, p, θ),
whereωBCCP = ωsat andωERTBP = 1, we look for functions
FM(q, p, θ) =∑n≥2
FMn (q, p, θ)
such that
{HM, FM} = 0.
This equation gives a recursive way of computing the approximatefirst
integralsFM. We solve it, for both models, up to order 16.
ERTBP BCCP
FM2 iq1p1 + iq2p2 iq1p1 + iq2p2 + iq3p3
19
BCCP: Zones of Effective Stability
Normal Form First Integral
(x-y section fort = 0) (x-y section fort = 0)
0.8669
0.86695
0.867
0.86705
0.8671
-0.4993 -0.49925 -0.4992 -0.49915 -0.4991 -0.49905 -0.499 -0.49895 -0.49890.86688
0.8669
0.86692
0.86694
0.86696
0.86698
0.867
0.86702
0.86704
0.86706
0.86708
0.8671
-0.4993 -0.49925 -0.4992 -0.49915 -0.4991 -0.49905 -0.499 -0.49895 -0.4989
(x-y projection)
0.8645
0.865
0.8655
0.866
0.8665
0.867
0.8675
-0.501 -0.5005 -0.5 -0.4995 -0.499 -0.4985 -0.498 -0.4975 -0.497
20
Quasi-Periodic Models
• TheBianular Problem(BAP).
• TheTricircular Coherent Problem(TCCP).
• A Preliminary Studyof theTCCP.
21
The Bianular Problem (I)
• We compute aquasi-periodicsolution, withtwo basic frequencies,
of the planarSun-Jupiter-SaturnThree Body Problem.
• This quasi-periodic solution lies on atorus. As the problem is
Hamiltonian, this torus belongs to afamily of tori.
• We look for a torus on this family for which the osculating
eccentricityof Jupiter’s orbit is quite well adjusted to the actual
one.
22
The Bianular Problem (II)
We split theconstructionof the model in four parts:
1. The reduced Hamiltonian of the planarThree Body Problem.
2. A methodfor computing 2-Dinvariant tori.
3. Finding the desired torus.
Continuationof the family of invariant curves.
4. TheHamiltonianof theBianular Problem.
23
The reduced Hamiltonian of the Three Body Problem
We take the Hamiltonian of the planarThree Body Problemwritten in theJacobicoordinates in a uniformlyrotatingreference frame and we make acanonical changeof variables (using theangular momentumfirst integral)in order to reduce this Hamiltonian from4 to 3 degrees of freedom.
H =1
2α
(P 2
1 +A2
Q21
)+
1
2β
(P 2
2 + P 23
)− K
−α
r− (1− µ)msat
r13− µmsat
r23
where
α = µ(1− µ)
β = msat/(1 + msat)
A = Q2P3 − Q3P2 + K
24
Computing 2-D invariant tori (I)
• The method can be found inCastella & Jorba, CelMech, 2000.
Basically, it is a method for computinginvariant curvesof maps.
• The map is choosen as thePoincare mapof the time-T flow
corresponding to the reduced Three Body Problem (let us call it
ΦT (·)), where the fixed timeT is the relative period of Saturn in the
Sun-Jupiter system(Tsat = 2π
ωsat
).
25
Computing 2-D invariant tori (II)
Let
ϕ(θ) = A0 +∑k>0
(Ak cos(kθ) +Bk sin(kθ))
be a parameterization of the invariant curve andω its rotation number.
Let, also,C(T1,Rn) be the space of continuous functions fromT1 in Rn,
and let us define the mapF : C(T1,Rn) → C(T1,Rn) as
F (ϕ)(θ) = ΦT (ϕ(θ))− (Tωϕ)(θ) ∀ϕ ∈ C(T1,Rn),
whereTω : C(T1,Rn) → C(T1,Rn) is the translation byω:
(Tωϕ)(θ) = ϕ(θ + ω).
It is clear that the zeros ofF in C(T1,Rn) correspond toinvariant curves
of rotation numberω.
26
Computing 2-D invariant tori (III)
As we do thecomputations numerically, we fix a truncation value for the
series (Nf ) and we construct adiscretizedversion of the functionF on
the following mesh of2Nf + 1 points onT1:
θj =2πj
2Nf + 1, 0 ≤ j ≤ 2Nf .
LetFNfbe this discretization ofF :
ΦT (ϕ(θj))− ϕ(θj + ω) , ∀0 ≤ j ≤ 2Nf .
So, given a set ofFouriercoefficientsA0,Ak andBk (1 ≤ k ≤ Nf ),
we can compute the pointsϕ(θj), thenΦT (ϕ(θj)) and next the points
ΦT (ϕ(θj))− ϕ(θj + ω), 0 ≤ j ≤ Nf .
27
Finding the desired torus (I)
For solving the equations, we use the well knownNewton method.
The initial approximationto the unknowns in the Newton method is given
by thelinearizationof the Poincare map around a fixed point (a periodic
orbit, for the flow)X0. We use theperiodic orbitcomputed in theBCCP
model.
X0 = ΦTsat(X0)
As there aretwo different non-neutralnormal directions, there will betwo
families of toristarting at the periodic orbit.
28
Finding the desired torus (II)
1. We compute a first torus for each family.
2. We add to the invariant curve equations the following one:
eccen(x1, x2, x3, x4, x5, x6,K) = e
whereeccen(·) is a function that gives usJupiter’s osculating
eccentricity(we evaluate it when Sun, Jupiter and Saturn are in a
particular collinear configuration); ande is a fixed constant that will
be used as acontrol parameter.
3. We makecontinuation of each familyincreasing the parametere to
its actual value. This is equivalent to use theangular momentumK
as parameter.
29
Finding the desired torus (III)
The results differ depending on the family.
• Family 1: As the parametere is increased, the number of harmonics
(Nf ) increases very much. We stop the continuation whenNf = 90(181 harmonics). The solution’s orbital elements are not as desired.
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
30
Finding the desired torus (IV)
• Family 2: Here, it is possible to continue the family of tori until
Jupiter’s eccentricityis e = 0.0484. The two frequencies of the
final torus are:
ω1 = 0.597039074021947 (Saturn’s frequency).
ω2 = 0.194113943490717 (ω1·ω2π ).
TheBianularSolution:
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
(Rotating Ref. Frame) (Inertial Ref. Frame)
31
The Hamiltonian of the BAP Model
It is possible to write theequationsof amassless particlethat moves under
the attraction of the three primaries. The correspondingHamiltonianis:
HBAP =1
2α1(θ1, θ2)(p
2x + p2
y + p2z) + α2(θ1, θ2)(xpx + ypy + zpz)
+α3(θ1, θ2)(ypx − xpy) + α4(θ1, θ2)x + α5(θ1, θ2)y
−α6(θ1, θ2)
[1− µ
qS+
µ
qJ+
msat
qsat
]
where
q2S = (x − µ)2 + y2 + z2, q2
J = (x − µ + 1)2 + y2 + z2,
q2sat = (x − α7(θ1, θ2))
2 + (y − α8(θ1, θ2))2 + z2,
θ1 = ω1t + θ(0)1 , θ2 = ω2t + θ
(0)2 .
32
The functionsα1(θ1, θ2) and α2(θ1, θ2)
α1(θ1, θ2) =∑
k≥(0,0)
α+1k cos(k1θ1 + k2θ2) +
∑k≥(1,1)
α−1k cos(k1θ1 − k2θ2)
α2(θ1, θ2) =∑
k>(0,0)
α+2k sin(k1θ1 + k2θ2) +
∑k≥(1,1)
α−2k sin(k1θ1 − k2θ2)
k1 k2 α+1k
α−1k
α+2k
α−2k
0 0 1.0012005708e+00
1 0 -2.4815098142e-04 7.3672422723e-05
0 1 1.4029937539e-04 -8.5639486462e-06
2 0 1.0960862810e-03 -6.4890366115e-04
1 1 -6.9015522178e-05 -1.7613232873e-06 2.9159795163e-05 -2.0823826150e-06
0 2 -8.1796687544e-07 3.7365269957e-07
3 0 1.0139637739e-04 -9.3726011699e-05
2 1 2.2645717633e-05 9.7074096718e-02 -1.4685465534e-05 -4.8448545618e-02
1 2 -3.7513125445e-05 -9.4041573332e-06 1.8541850579e-05 6.3099860119e-07
0 3 -4.9135929446e-08 2.3959571832e-08
4 0 3.1920890353e-05 -3.6373210600e-05
3 1 4.7465358994e-07 -4.2502882354e-05 -1.8465965247e-06 2.4285825409e-05
2 2 6.6876937521e-07 -1.3940506628e-05 -4.5339192162e-07 8.5955542933e-06
1 3 4.3128955834e-08 -5.8373958936e-07 -2.1231004810e-08 -8.7460026101e-09
0 4 7.0718574586e-10 1.0155648952e-10
33
The Tricircular Coherent Problem
• We compute aquasi-periodicsolution, withtwo basic frequencies, of
the planarSun-Jupiter-Saturn-UranusFour Body Problem. We use
theJacobi coordinates.
• We take the frequencies ofSaturnandUranusin the rotating
Sun-Jupiter system.
• Finally, we write theHamiltonianof a fifth masslessparticlethat
moves under the action of the four primaries, supposing that they are
following the solution found before.
34
The Jacobi coordinates
r
R
z
SUN
JUPITER
SATURN
URANUS
35
The Hamiltonian of the SJSU problem
We take the Hamiltonian of theplanar 4BPwritten in theJacobicoordinates in a
uniformly rotatingreference frame and we make a canonical change of variables
(using theangular momentumfirst integral) in order to reduce thisHamiltonian
from 6 to 5 degrees of freedom.
H =1
2α
(P 2
1 +A2
Q21
)+
1
2β
(P 2
2 + P 23
)+
1
2γ
(P 2
4 + P 25
)− K
−α
r− (1− µ)msat
r13− µmsat
r23
− (1− µ)mura
r14− µmura
r24− msatmura
r34
whereα = µ(1− µ) γ = (1+msat)mura
1+msat+mura
β = msat/(1 + msat) A = Q2P3 − Q3P2 + Q4P5 − Q5P4 + K
36
Computation of a first torus (I)
First, we look for aquasi-periodic solutionof SJSU equations formura = 0. We use the same method as before. That is, to computeaninvariant curveϕ(·) of the mapΦT (·):
ΦT (ϕ(θ)) = ϕ(θ + ω) , ∀θ ∈ T.
As a first approximation for theNewton method, we use the previouslycomputedperiodic orbitfor SJS and theKeplerianorbit for Uranus.
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
37
Computation of a first torus (II)
The angle between the initial and final position ofUranusin the last
trajectory is very close to:
ω =2πωura
ωsat(mod2π) = 2.750807556.
Thus, if we keep constant the valueTsat and if we imposeω to be the
rotation numberof the invariant curve, thefrequenciesof the 2-D
invariant torus that we are computing will be:
• ωsat (Saturn’s relative frequency in the Sun-Jupiter system)
• ωura (Uranus’ relative frequency in the Sun-Jupiter system).
38
The Tricircular Coherent Solution of the SJSU Problem
Once a solution formura = 0 is computed, by means of acontinuation
method we proceed to increase the parametermura up to its actual value,
mantaining the two internalfrequenciesof the torus.
-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
-4 -3 -2 -1 0 1 2 3 4
(Rotating Ref. Frame) (Inertial Ref. Frame)
39
The Hamiltonian of the TCCP Model
Finally, it is possible to write the equations of amassless particlethatmoves under the attraction of thefour primaries. The correspondingHamiltonianis:
HTCCP =1
2α1(θ1, θ2)(p
2x + p2
y + p2z) + α2(θ1, θ2)(xpx + ypy + zpz)
+α3(θ1, θ2)(ypx − xpy) + α4(θ1, θ2)x + α5(θ1, θ2)y
−α6(θ1, θ2)
[1− µ
qS+
µ
qJ+
msat
qsat+
mura
qura
]where
q2S = (x− µ)2 + y2 + z2, q2
J = (x− µ + 1)2 + y2 + z2,
q2sat = (x− α7(θ1, θ2))2 + (y − α8(θ1, θ2))2 + z2,
q2ura = (x− α9(θ1, θ2))2 + (y − α10(θ1, θ2))2 + z2,
θ1 = ωsatt + θ(0)1 , θ2 = ωurat + θ
(0)2 .
40
Preliminary Study of the TCCP
• Local linear analysisnear the triangular points.
– Symplecticquasi-periodic FloquetTransformation.
• High-ordernormal form.
– Non-lineardynamics.
• Approximatefirst integral.
– EffectiveStability.
41
Local Study aroundL5
Jorba & Simo, 96:
In theTCCPsystem, theL5 point is replaced by a2-D invariant torus: T5.
T5
0.8645
0.865
0.8655
0.866
0.8665
0.867
0.8675
-0.501 -0.5005 -0.5 -0.4995 -0.499 -0.4985 -0.498 -0.4975 -0.497-0.5005
-0.5
-0.4995
-0.499
-0.4985
-0.498
-0.4975
-0.8675 -0.867 -0.8665 -0.866 -0.8655 -0.865 -0.8645
(x-y projection) (px-py projection)
We will see that the linearnormal modesof T5 have modulus exactly1
=⇒ T5 is Linearly Stable
42
Symplectic Quasi-Periodic Floquet Transformation (I)
Thelinear flowaround the 2-D invariant torusT5:
z = Q(θ1, θ2)z
θ1 = ωsat
θ2 = ωura
(θ1 = 2π)-Poincaresection=⇒ Linear quasi-periodicskew product:
z = A(θ)z
θ = θ + ω
whereω = 2π(
ωura
ωsat− 1
)= 2.75080755611202.
43
Symplectic Quasi-Periodic Floquet Transformation (II)
Using the method byJorba 2001, we reduce thisquasi-periodicskewproduct to
y = Λy,
by implementing a linear change of variablesz = C(θ)y.
Λ:
• Diagonal complex6× 6 matrix.
• Constantcoefficients.
• A(θ)C(θ) = C(θ + ω)Λ.
If we define
• Tω : Ψ(θ) ∈ C(T1,Cn) → Ψ(θ + ω) ∈ C(T1,Cn).
• Ψj(θ): j-th column of the matrixC(θ).we obtain ageneralized eigenvalueproblem:
A(θ)Ψj(θ) = λjTωΨj(θ).
44
Symplectic Quasi-Periodic Floquet Transformation (III)
Complex change:
z = Q(θ1, θ2)zz=P c(θ1,θ2)y−−−−−−−−−→
(1)y = DByy y
z = A(θ)zz=C(θ)y−−−−−−→
(2)y = Λy
(1) DB = diag(iω1, iω2, iω3,−iω1,−iω2,−iω3)
ωj is such thatλj = exp(iωjTsat)
(2) P c(θ1, θ2) = Q(θ1, θ2)Pc(θ1, θ2)− P c(θ1, θ2)DB ; P c(t = 0) = C(θ
(0)2 )
(3) A(θ)C(θ) = C(θ + ω)Λ
Real change: z = Q(θ1, θ2)zz=P r(θ1,θ2)y−−−−−−−−−→
(3)x = Bx
(3) P r(θ1, θ2) = Q(θ1, θ2)Pr(θ1, θ2)− P r(θ1, θ2)B ; P r(t = 0) = R(θ
(0)2 )
45
Symplectic Quasi-Periodic Floquet Transformation (IV)
Matrices:
R(θ) =1
2C(θ)
I3 −iI3
I3 iI3
B = R−1CDBC−1R =
0 0 0 ω1 0 0
0 0 0 0 ω2 0
0 0 0 0 0 ω3
−ω1 0 0 0 0 0
0 −ω2 0 0 0 0
0 0 −ω3 0 0 0
P r(θ1, θ2) is asymplecticmatrix.
Thesecond degreeterms of the Hamiltonian:
Hr2 (x, y) =
1
2ω1(x
21 + y2
1) +1
2ω2(x
22 + y2
2) +1
2ω3(x
23 + y2
3)
where thefrequenciesareω1 = − 0.080473064872369,
ω2 = 0.996680625156409 andω3 = 1.00006269133083.
46
High-Order Normal Form
• Expansionof the Hamiltonian by means of theLegendrepolynomials
recurrence.
• Normalizingprocedure with theLie series method implemented as in
Jorba ’99.
H = 〈$, pθ〉+N (q1p1, q2p2, q3p3) +R(q1, q2, q3, p1, p2, p3, θ1, θ2)
In realaction-anglecoordinates,N does not depend on the anglesϕj but
only on the actionsIj :
N =[N/2]∑|k|=1
hkIk11 Ik2
2 Ik33 , k ∈ Z3, hk ∈ R
47
Non-Linear Dynamics (I)
0.864
0.8645
0.865
0.8655
0.866
0.8665
0.867
0.8675
-0.501 -0.5005 -0.5 -0.4995 -0.499 -0.4985 -0.498 -0.4975 -0.497-0.5005
-0.5
-0.4995
-0.499
-0.4985
-0.498
-0.4975
-0.8675 -0.867 -0.8665 -0.866 -0.8655 -0.865 -0.8645 -0.864
-0.501-0.5005 -0.5 -0.4995-0.499-0.4985-0.498-0.4975-0.497 0.864 0.8645
0.865 0.8655
0.866 0.8665
0.867 0.8675
-0.04-0.03-0.02-0.01
0 0.01 0.02 0.03 0.04
48
Non-Linear Dynamics (II)
0.85
0.855
0.86
0.865
0.87
0.875
0.88
0.885
-0.52 -0.515 -0.51 -0.505 -0.5 -0.495 -0.49 -0.485 -0.48 -0.475-0.515
-0.51
-0.505
-0.5
-0.495
-0.49
-0.485
-0.88 -0.875 -0.87 -0.865 -0.86 -0.855 -0.85
-0.52-0.515-0.51-0.505 -0.5 -0.495-0.49-0.485-0.48-0.475 0.85 0.855
0.86 0.865
0.87 0.875
0.88 0.885
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
-0.88-0.875
-0.87-0.865
-0.86-0.855
-0.85-0.515-0.51
-0.505-0.5
-0.495-0.49
-0.485
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.9
-0.55 -0.54 -0.53 -0.52 -0.51 -0.5 -0.49 -0.48 -0.47 -0.46 -0.45-0.015
-0.01
-0.005
0
0.005
0.01
0.015
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
49
Approximate First Integral
{H, F} = 0
⇓
F (q, p, θ1, θ2) = i(q1p1 + q2p2 + q3p3) +
N∑n=3
∑|k|=n
∑j=(j1,j2)
(f
kn,je
i(j1θ1+j2θ2))
qk1
pk2
wherefkn,j =
ickn,j
j1ωsat+j2ωura−〈k2−k1,ω〉 andckn,j are known.
Zone ofEffective Stability:
0.8645
0.865
0.8655
0.866
0.8665
0.867
0.8675
-0.501 -0.5005 -0.5 -0.4995 -0.499 -0.4985 -0.498 -0.4975 -0.497
50
Conclusions
• First Stepof aLong-term projectwhichUltimate Goalis:To compute an approximate
quasi-periodic solution for a
Trojanasteroid.
−→To conclude its practicalstabil-
ity for the estimated life time of
theSolar system.
• Models:
RTBP −→ BCCP −→ TCCP
↓ ↓
ERTBP −→ BAP
• Numerical methods
andsoftware:
– Quasi-periodicsolutions.
– Symplectic Quasi-PeriodicFloquetTheory.
– High-OrderNormal Forms.
– ApproximateFirst Integrals.
– Regions ofEffective Stability.
51
