Stability of stationary points of a body under the ...web.abo.fi/fak/mnf/mate/CADE2007/Talks/Elipe1.pdf · Stability of stationary points of a body under the gravitational ﬁeld

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.


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.


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


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?


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


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 ⇒


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


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)).


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

,


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,


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.


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?


-3 -2 -1 1

-3

-2

-1

1

α

β


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.


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


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


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


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)]

,


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))


Case D4 = 0

Points (α, β) such that D4 = 0,

-3 -2 -1 1

-3

-2

-1

1

α

β


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


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


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


(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


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


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.


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.


-3 -2 -1 1

1.5

1.75

2.25

2.5

2.75

α

|b2000|/|b0010|

STABLE


-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


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

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Stability of stationary points of a body under the ...web.abo.fi/fak/mnf/mate/CADE2007/Talks/Elipe1.pdf · Stability of stationary points of a body under the gravitational ﬁeld