Stability of Periodic Motions in Satellite Dynamics - Stability ...Stability of Hamiltonian systems with more then two degrees of freedom Stability of Relative Equilibrium Positions

Stability of Hamiltonian systems with more then two degrees of freedomStability of Relative Equilibrium Positions of Satellite in a Circular Orbit

Stability of Periodic Motions in SatelliteDynamics

Stability Theory for Hamiltonian Systems

Boris S. Bardin

Faculty of Applied Mathematics and PhysicsDepartment of Theoretical Mechanics

Course of Computer Algebra and Differential Equations

Boris Bardin


Contents

1 Stability of Hamiltonian systems with more then two degreesof freedom

Stability for most initial conditionsFormal stability

2 Stability of Relative Equilibrium Positions of Satellite in aCircular Orbit

Hamiltonian of the Perturbed MotionLinear Problem of StabilityNonlinear Problem of Stability

Boris Bardin



Contents





Boris Bardin



Arnold’s Theorem of Stability for most of initialconditions

Let us consider a Hamiltonian system of k degrees of freedom.The Birkhoff normal form of the Hamiltonian

H = H(0)(r) + H(1)(r,ϕ), (1)

where

H(0)(r) = ω1r1 + · · ·+ ωk rk +k∑

i,j=1

aij ri rj ,

and H(1)(r,ϕ) are terms of the order higher then two withrespect ri .

Boris Bardin



Arnold’s Theorem of Stability for most of initialconditions

If at r1 = · · · = rk = 0 Hamiltonian (1) satisfy the condition ofnondegeneracy

Dk = det(∂2H0∂r2

)6= 0 (2)

or the condition of isoenergetic nondegeneracy

Dk+1 = det

(∂2H0∂r2

) (∂H0∂r

)(∂H0∂r

)0

6= 0, (3)then the Hamiltonian system is stable for most of initialconditions (in the sense of Lebesque measure).

Boris Bardin



Arnold diffusion

Conditions (7) and (8) are not the same.

Example: H(0) = ω1r1 − ω2r2 + (r1 + r2)2

D2 = 0, D3 = −2(ω1 + ω2)2 6= 0.

The stable for most of initial conditions system can be unstablein the sense of Liapunov. Such a phenomenon is called Arnolddiffusion.

The typical case in higher-dimensional problems is topologicalinstability.Arnold’s example (Dokl. Acad. Nauk SSSR, V.156, 1964, 9-12):

H =12(I21 + I

22) + ε(cosϕ1 − 1)(1 + µ(sinϕ2 + cos t))

Boris Bardin



Nehoroshev’s Theorem

The KAM theory proves the metric stability, i.e stability for mostof initial conditions

Nehoroshev’s TheoremSuppose that the unperturbed Hamiltonian H(0)(I) is a steepfunction. Then for a sufficiently small perturbation

|I(t)− I(0)| < εb for 0 ≤ t ≤ 1ε

exp(

1εa

)in the perturbed Hamiltonian system. Here a and b are positiveconstants that depend on the geometric properties of theunperturbed Hamiltonian.

Boris Bardin



Arnold diffusion in resonant casesMarkeev’s Example

Let us consider the system with Hamiltonian

H = ω1r1 − ω2r2 + ω3r3 + r1r3 − r1r2 + r2r3 + H(1)(r,ϕ),

whereH(1)(r,ϕ) = r1r2

√r3 sin(2ϕ1 + 2ϕ2 + ϕ3).

and resonance 2ω1 − 2ω2 + ω3 = 0 takes placeThe conditions of Arnold’s theorem are fulfilled

D4 = (ω1 + ω2)2 6= 0,

but the Hamiltonian system has the following solution

2ϕ1 + 2ϕ2 + ϕ3 = π

r3(t) =12

r1(t) =12

r2(t) = r3(0)[1− 6r32

3 (0)t ].

Boris Bardin



Contents





Boris Bardin



Formal stability

Definition: If there exist power series

G(q,p) = G2(q,p) + G3(q,p) + . . . ,

such that it is a formal positive definite integral of theHamiltonian system

dqdt

=∂H∂p

,dpdt

= −∂H∂q

, (4)

then system (4) is called formal stable.

Boris Bardin



Moser’s Theorem on Formal stability

Let us consider a Hamiltonian system of k degrees of freedomwith the Hamiltonian.

H = ω1r1 + · · ·+ ωk rk + H(1)(r,ϕ), (5)

here H(1)(r,ϕ) are terms of the order higher then one withrespect ri .If the following conditions

n∑i=1

miωi 6= 0, mi ∈ Z, mi ≥ 0,n∑

i=1

mi > 0

are fulfilled, then the system with Hamiltonian (5) is formalstable.(Moser, J. New aspects in the theory of stability of Hamiltoniansystems. Comm. Pure Appl. Math. V.11, 1958, 81–114.)

Boris Bardin



Glimm’s Theorem on Formal stability

Let us suppose that the normal form of the Hamiltonian reads

H = ω1r1 + · · ·+ ωk rk +k∑

i,j=1

aij ri rj + H(1)(r,ϕ), (6)

andn∑

i=1

miωi 6= 0, mi ∈ N, 0 <n∑

i=1

|mi | ≤ 4.

If the quadratic formk∑

i,j=1

aij ri rj is definite, then the system with

Hamiltonian (6) is formal stable.(Glimm, J. Formal stability of Hamiltonian systems. Comm.Pure Appl. Math. V.17, 1964, pp. 509–526.)

Boris Bardin



Brjuno’s Theorem on Formal stability

If the quadratic formk∑

i,j=1

aij Ii Ij 6= 0,

for the vector I = (I1, . . . , Ik ) 6= 0 such that Ii ≥ 0 and Ii aresolutions of the equation

n∑i=1

mi Ii = 0,

then the system with Hamiltonian (6) is formal stable.

(Brjuno, A. D. Formal stability of Hamiltonian systems.(Russian) Mat. Zametki V.1, 1967, pp. 325–330.)

Boris Bardin



Contents





Boris Bardin



Relative Equilibrium Positions of Satellite in a CircularOrbit

Orbit of the mass center n

v

Boris Bardin



The Hamiltonian of the Problem

H =(sin2 ϕ+ θA cos2 ϕ)

2 sin2 ϑp2ψ +

[(sin2 ϕ+ θA cos2 ϕ) +

θAθC

]p2ϕ2

(cos2 ϕ+ θA sin2 ϕ)p2ϑ2− (sin2 ϕ+ θA cos2 ϕ)pψpϕ+

(1− θA) sin 2ϕ2 sinϑ

pψpϑ −(1− θA)

2sin 2ϕ cotϑpψpϕ − pψ

32

[θA − 1θA

a211 +θC − 1θA

a213

]

where

a11 = cosψ cosϕ− sinψ sinϕ cosϑ, a13 = sinψ sinϑ

Boris Bardin



Variations

Parameters: θA = A/B and θC = C/BEquilibrium position:

ψ = ϕ = pϑ = pψ = 0, ϑ =π

2, pψ =

1θA

Let us introduce variations

q1 = ϑ−π

2, q2 = ϕ, q3 = ψ,

p1 = pϑ, p2 = pϕ, p3 = pψ −1θA

Boris Bardin



Perturbed Hamiltonian

H = H2 + H3 + H4 + . . .

H2 =p212

+θA

2θCp22 +

12θA

q21 −(θA − 1)(3θA + 1)

2θ2Aq23+

+ q1p2 −θA − 1θA

q2p1 +θA2

p23 +3(θC − θA)

2θA

H3 = q21p3 +3(θA − 1)

θAq1q2q3 + θAq1q2p3 −

3θA − 1)θA

q22q3−

− (θA − 1)q2p1p3

Boris Bardin



Contents





Boris Bardin



Linear Problem of StabilitySufficient condition of stability

If the following condition is fulfilled

θA < θC < 1

then relative equilibrium is stable in Lyapunov sense.

Sufficient condition of stability: If the biggest principal axis isdirected along the radius vector of the center of mass and thesmallest one is directed along the normal of the orbit.

Boris Bardin



Boris Bardin



Linear Problem of StabilityNecessary condition of stability

If parameter values belong to the Domain I then relativeequilibrium is stable in linear approximation.In domain II

H =12ω1(q21 +p

21)−

12ω2(q22 +p

22)+

12ω2(q23 +p

23)+H3+H4+ . . .

where ω1 > 0, ω2 > 0, ω3 > 0. Nonlinear study isnecessary to solve stability problem.

Boris Bardin



Contents





Boris Bardin



Nonlinear Problem of StabilityNonresonant case

In polar variables: qi =√

2ri sinϕi , pi =√

2ri cosϕi

H = H(0)(r1, r2, r3) + H̃(r1, r2, r3, ϕ1, ϕ2, ϕ3)

where

H(0) = ω1r1 − ω2r2 + ω3r3++ c200r21 + c110r1r2 + c101r1r3 + c020r

22 + c101r1r3 + c002r

23

Boris Bardin



Stability for most of initial conditions

In Domain II outside of curves of resonances up to fourth orderat r1 = r2 = r3 = 0 Hamiltonian (1) satisfy the condition ofnondegeneracy

D3 = det(∂2H0∂r2

)6= 0 (7)

or the condition of isoenergetic nondegeneracy

D4 = det

(∂2H0∂r2

) (∂H0∂r

)(∂H0∂r

)0

6= 0, (8)Thus, the Hamiltonian system is stable for most of initialconditions (in the sense of Lebesque measure).

Boris Bardin



Formal Stability

If in the domain ri ≤ 0 (i = 1,2,3) the system of equations

ω1r1 − ω2r2 + ω3r3 = 0c200r21 + c110r1r2 + c101r1r3 + c020r

22 + c101r1r3 + c002r

23 = 0

does not have any solution except trivial one r1 = r2 = r3 = 0,then the equilibrium is formal stable.

Boris Bardin



Boris Bardin



Resonances

In the considered approximation the following resonances cantake place

k1ω1 − k2ω2 + k3ω3 = 0 (|k1|+ |k2|+ |k3| = 2,3,4)

List of resonances which should be taken into account:

1. ω1 − ω2 − 2ω3 = 0, 2. ω1 − ω2 − ω3 = 0,

3. ω2 − ω3 = 0, 4. ω1 + ω2 − 2ω3 = 0,

5. ω1 − ω3 = 0, 6. 2ω2 − ω3 = 0,

7. ω1 + ω2 − ω3 = 0, 8. ω1 − 3ω2 = 0,

9. 2ω1 − ω3 = 0,

Boris Bardin



Boris Bardin



Arnold Diffusion

In points Pi (i = 1,2,3) of the Domain II the fifth orderresonance 4ω2 = ω1 takes place.Equality 16c020 + 4c011 + c002 = 0 is fulfilled

The Hamiltonian normal form reads

H = H(0) + γi r22√

r3 cos(4ϕ2 + ϕ3) + O((r1 + r2 + r4)3)

The truncated system has the following partial solution

r1 = 0, r2 = 4r3 > 0.

Boris Bardin



Arnold Diffusion

Boris Bardin

Stability of Hamiltonian systems with more then two degrees of freedomStability for most initial conditionsFormal stability

Stability of Relative Equilibrium Positions of Satellite in a Circular OrbitHamiltonian of the Perturbed MotionLinear Problem of StabilityNonlinear Problem of Stability

Stability of Periodic Motions in Satellite Dynamics - Stability ...Stability of Hamiltonian systems with more then two degrees of freedom Stability of Relative Equilibrium Positions

Documents