stability of stationary points of a body under the...
TRANSCRIPT
Stability of stationary points of a body
under the gravitational field of a finite body
Antonio Elipe
Grupo de Mecanica Espacial, University of Zaragoza, Spain
Vıctor Lanchares
Dpt. Maths & Computation, University of La Rioja, Spain
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 2
Artificial Satellite x = F (x, x, t), x ∈ R3
In general,
x = F (x, t) = −∇xU,
with U a Series expansion; e.g. in spherical coordinates
U = −µr
1 +∑
n≥2
(α
r
)n
JnPn(cosβ)
+∑
1≤m≤n(Cm
n cosmλ + Smn sinmλ)Pmn (cos β)
where
Pmn Legendre Polynomials.
Jn zonal harmonics; Cmn , S
mn tesseral harmonics.
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λ = λ(t), hence, t appears explicitly !
Alternative:
Formulate the problem in a synodic frame.
Consequences:
1) t no longer appears.
2) There is a new term in the kinetic energy
−wΩ
which may cause difficulties in the tesseral case.
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For axial symmetry, we have the Zonal Problem
U = −µr
1 +∑
n≥2
(α
r
)nJnPn(cos β)
.
J2 ≈ 10−3, Jn < 10−6, α≪ r. Thus,
U = U0 + U1 + U2 + . . . , con U0 ≫ U1 ≫ U2 ≫ . . .
If U = U0 = −µr, Kepler Problem
When U = U0 + U1 = −µr
1 +(α
r
)2J2P2(cos β)
,
The Main Problem Non integrable !
Perturbed Kepler Problems
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 5
Geostationary points (equilibria)
Zonal problem.- Circle of radius 42 624 km
Tesseral problem.- 4 equilibria
- two stable (linear and Lyapunov)
- two unstable
Stationary points of an oblate planet (attracting body)
How many equilibria are?
What about the stability?
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 6
Oblate planet
Tesseral Problem up to J22
Synodic reference frame rotating with the planet (ω)
Origin at the center of mass of the planet
Axes coincide with the principal axes of inertia
V = −µr
1 +
⊕r
2
3 Γ2,2x2 − y2
r2− 1
2Γ2,0
1 − 3z2
r2
,
Γ2,0 = C2,0, Γ2,2 =√
C22,2 + S2
2,2, Γ2,0 < 0 < Γ2,2
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 7
Lagrangian
L = 12(x2 + y2 + z2) + ω(xx− yy) + U(x, y, z),
where U is the effective potential function
U = 12ω
2(x2 + y2) − V(x, y, z),
Equations of motion
x− 2ω y = Ux,
y + 2ω x = Uy,
z = Uz.
Equilibria:
Ux = Uy = Uz = 0 ⇒
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 8
z = 0 and
(+) y = 0
(−) x = 0
r
ak
5
−
r
ak
2
= 3(−12C2,0 ± 3C2,2)
α
ak
2
r
ak= 1+ǫ−3ǫ2+
44
3ǫ3−260
3ǫ4+567ǫ5−35581
9ǫ6+
259160
9ǫ7+O(ǫ)8
ǫ = (−12C2,0 ± 3C2,2)
α
ak
2
(+) y = 0
(−) x = 0
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Possible solutions
In general, there are 6 solutions:
E1(±r1, 0), E2(0,±r2),
and
E′2(0,±r′2)
For the Earth, only 4 equilibria (E1(±r1, 0)) and (E2(0,±r2)).
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 10
Linear stability
Equilibrium at the origin
x = η1 + r1, X = H1, y = −ξ1, Y = ωr1 − Ξ1
x = ξ2, X = Ξ2 − ωr2, y = η2 + r2, Y = H2,
(for E′2 we change r2 by r′2).
The transformed Hamiltonians are
H(j) = −1
2ω2r2
j +1
2(Ξ2
j +H2j ) − ω(ξjHj − ηjΞj) − ω2rjηj
− µ
ρj
1 − ⊕2
ρ2j
1
2Γ20 − (−1)i3 Γ22
ξ2j − η2
j − 2rjηj − r2j
ρ2j
,
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Expansion by Taylor series
H(j) = H(j)2 + H(j)
3 + H(j)4 + . . . .
H(j)n is a homogeneous polynomial of degree n in the new variables.
H(j)2 =
1
2(Ξ2
j+H2j )−ω(ξjHj−ηjΞj)+
1
2ω2(αj ξ
2j+βj η
2j ) =
1
2ζjAjζj,
the parameters αj, βj are
αj = 1 − 12 (−1)jΓ22a3k⊕2
r5j
and βj = 2
a3k
r3j
− 2
.
Equations of motion
ζ = JAζ = Bζ,
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Linear stability Eigenvalues are pure imaginary
B =
0 ω 1 0
−ω 0 0 1
−ω2α 0 0 ω
0 −ω2β −ω 0
.
Characteristic equation
det(λI − B) = λ4 + ω2(α + β + 2)λ2 + ω4(1 − α)(1 − β) = 0,
Eigenvalues
λ2j,1 =
ω2
2
(
−(α + β + 2) +√
(α− β)2 + 8(α + β))
< 0,
λ2j,2 =
ω2
2
(
−(α + β + 2) −√
(α− β)2 + 8(α + β))
< 0.
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 13
RI = (α, β) ∈ R2 |α, β > 1, and
RII = (α, β) ∈ R2 | − 3 < α, β < 1,
and (α− β)2 + 8(α + β) > 0.
E1(±r1, 0) UNSTABLE
E′2(0,±r′2) UNSTABLE
E2(0,±r2) LINEARLY STABLE, Lyapunov stable?
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 14
-3 -2 -1 1
-3
-2
-1
1
α
β
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Theorem 1 (Arnold) Consider a two degrees of freedom Hamiltonian system H ex-
pressed, in real canonical coordinates (Φ1,Φ2, φ1, φ2), as
H = H2 + H4 + . . .+ H2n + H,
where:
1. H is real analytic in a neighborhood of the origin R4,
2. H2k, 1 ≤ k ≤ n, is a homogeneous polynomial of degree k in Φi, with real coefficients.
In particular,
H2 = ω1Φ1 − ω2Φ2, 0 < ω1, 0 < ω2;
H4 =1
2(AΦ2
1− 2BΦ1Φ2 + CΦ2
2) .
3. H has a power expansion in Φi which starts with terms at least of order 2n+ 1.
Under these assumptions, the origin is a stable equilibrium provided for some k, 2 ≤k < n, H2 does not divide H2k or equivalently, provided D2k = H2k(ω2, ω1) 6= 0 and for
2 ≤ j < k, D2j = H2j(ω2, ω1) = 0.
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 16
Action and angle variables
Canonical transformation to complex variables
w = (u, v, U, V ) 7−→ ζ = (ξ, η,Ξ, H)
by the linear transformation
B =
ia1 −ia2 a1 a2
−b1 b2 −ib1 −ib2
b1ω − a1ω1 a2ω2 − b2ω −i(a1ω1 − b1ω) −i(a2ω2 − b2ω)
i(a1ω − b1ω1) −i(a2ω − b2ω2) a1ω − b1ω1 a2ω − b2ω2
,
ai, bi are functions of the frequencies ωi =√
λ2i
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 17
H2 = i ω1uU + i ω2v V,
Poincare variables (φ1, φ2, I1, I2)
u =√I1 exp(i φ1), v =
√I2 exp(−i φ2),
U = −i√I1 exp(−i φ1), V = i
√I2 exp(i φ2),
H2 = ω1I1 − ω2I2
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 18
H2 = i ω1uU + i ω2v V,
Normalization The Lie derivative L2 : F → (F,H2)
L2 = i ω1
u∂
∂u− U
∂
∂U
+ i ω2
v∂
∂v− V
∂
∂V
.
In the algebra of homogeneous polynomials in (u, v, U, V ),
L2 (umUnvpV q) = [i ω1 (m− n) + i ω2 (p− q)] umUnvpV q;
The kerL2 is generated by monomials of the type (uU)m(v V )p
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 19
Normalized Hamiltonian (Poincare variables)
ν#H = ω1I1 − ω2I2 + AI21 − 2B I1I2 + C I2
2 + H6 + . . . ,
D4 = H4(ω2, ω1) = Aω22 − 2B ω1ω2 + C ω2
1.
D4 =
N
24 (1 − β)2D
ω2
r22
.
The coefficients N and D are the polynomials
N =∑
0≤m≤5
∑
0≤n≤7−mAm,n α
mβn;
D =[
(α− β)2 + 8(α + β)] [
4 (α− β)2 − 9(1 + α β) + 41(α + β)]
,
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 20
0.002 0.006 0.01
-75
-25
25
75
D 4
rE2 Γ
22
D4 6= 0 then, STABLE except:
Resonances 1:1, 2:1,
D4 = 0 (⊕2 Γ22 = 4.3862 × 10−3 (Γ22 = 0.192385))
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Case D4 = 0
Points (α, β) such that D4 = 0,
-3 -2 -1 1
-3
-2
-1
1
α
β
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 22
Case D4 = 0
At each point (α, β) such that D4 = 0,
– we go further with the normalization
– we obtain H6(Φ,Ψ)
– we compute D6 = H6(ω2, ω1). D6 > 1 then, STABLE
-2.5 -2 -1.5 -1 -0.5 0.5 1
10
20
30
40
logD
6
α
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 23
Resonances
ω1 = Ωm, ω2 = Ωn with Ω a frequency
Resonances introduce Zero Divisors in the normalization.
Extended Lissajous variables (ψ1, ψ2,Ψ1,Ψ2) avoid ZD
Unperturbed Hamiltonian H2 = ΩΨ2
Lie derivative L2(F ) =∂F
∂ψ2
F (ψ1, ψ2,Ψ1,Ψ2) ∈ kerL2 ⇐⇒ ∂F
∂ψ2= 0
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 24
M1 = 12Ψ1, S = 2−(m+n)/2(Ψ1 − Ψ2)
m/2(Ψ1 + Ψ2)n/2 sin 2mnψ1,
M2 = 12Ψ2, C = 2−(m+n)/2(Ψ1 − Ψ2)
m/2(Ψ1 + Ψ2)n/2 cos 2mnψ1,
with the constraint
C2 + S2 = (M1 +M2)n(M1 −M2)
m,
The normalized Hamiltonian is
Hs =∑
2(a1+a2)+(m+n)(a3+a4)=saa1a2a3a4
Ma1
1 Ma2
2 Ca3Sa4
The phase flow is the intersection on both surfaces
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 25
(I1, I2) 7→ (ω2, ω1) ⇐⇒M2 7→ 0
-20
2 -20
2
0
0.5
1
1.5
2-2
02 C1
-0.050.05
S1
1
1.1
M1
C1
1
1.1
M1
If both surfaces cut each other trnasversally, them Unstable
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 26
Resonance 2:1 ω1 = 2ω2 = 2Ω
H3 = a0010C + a0001S,
C2 + S2 = M 31 ,
=⇒H3 = b0010C,
C2 = M 31 ,
0.9 0.92 0.94 0.96 0.98
-0.1
-0.05
0.05
0.1
0.15
b0010
α
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 27
b0010 6= 0 except for α = 0.982359, then, the origin is
unstable elsewhere, except for this value of α
How is the stability for this value?
⋆ Push forward the normalization for this value
⋆ Compute the surface for M2 = 0
⋆ H4 = 1.42328C2, then, stable.
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 28
Resonance 3:1 ω1 = 3ω2 = 3Ω
H4 = b2000M21 + b0010C
C2 = M 41
and at the origin, they are
C = −b2000
b0010M 2
1
C = ±M 21
=⇒
=⇒If |b0010| < |b2000| 7→6 ∃ cuts, then stable.
If |b0010| > |b2000| 7→ ∃ cuts, then unstable.
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 29
-3 -2 -1 1
1.5
1.75
2.25
2.5
2.75
α
|b2000|/|b0010|
STABLE
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 30
-3 -2 -1 1
-3
-2
-1
1
α
β
2:1
3:1
-0.4 -0.2 0.2 0.4 0.6 0.8 1
-0.4
-0.2
0.2
0.4
0.6
0.8
1
α
β
D4 = 0
Antonio Elipe & Vıctor Lanchares CADE 2007. Turku. 31
Conclusions
• Lyapunov stability has been analyzed for a type of Hamiltonian
depending on two parameters
• Stationary satellites belong to this type of Hamiltonians
• Resonances among the natural frequencies have been consid-
ered
• The study is analytical; thus for a specific planet we only need
to replace the values of the parameters to check the Lyapunov
stability