Stability of a Hamiltonian System in a Limiting Case

c Pleiades Publishing, Ltd., 2012. ISSN 1560-3547, Regular and Chaotic Dynamics, 2012, Vol. 17, No. 1, pp. 24–35. 

Stability of a Hamiltonian System in a Limiting Case Kenneth R. Meyer1* , Jes´ us F. Palaci´ an2** , and Patricia Yanguas2*** 1

Department of Mathematical Sciences University of Cincinnati Old Chem 839, Cincinnati, 45221-0025 Ohio, USA 2 Departamento de Ingenier´ıa Matem´ atica e Inform´ atica Universidad P´ ublica de Navarra 31006 Pamplona, Spain Received September 23, 2011; accepted November 9, 2011

Abstract—We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov stable in a degenerate case. That is the 1 : −1 resonance case where the linearized system has double pure imaginary eigenvalues ±iω, ω = 0 and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point is encased in invariant tori and thus it is stable. This result implies the stability of the Lagrange equilateral triangle libration points, L4 and L5 , in the planar circular restricted three-body problem when the mass ratio parameter is equal to μR , the critical value of Routh. MSC2010 numbers: 34C20, 34C25, 37J40, 70F10, 70K65 DOI: 10.1134/S1560354712010030 Keywords: stability, Lagrange equilateral triangle, KAM tori, periodic solutions, planar circular restricted three-body problems, Routh’s critical mass ratio

To the memory of Jacques Henrard 1. INTRODUCTION For us the prototypical question is the stability of the Lagrange equilateral triangle libration points, L4 and L5 , in the planar circular restricted three-body problem when the mass ratio parameter is equal to μR , the critical value of Routh. The planar circular restricted three-body problem is a two degrees of freedom Hamiltonian system with a parameter, the mass ratio parameter μ, which sweeps the equilibrium at L4 , equivalently at L5 , through many resonances giving rise to numerous bifurcation results and stability analysis. In the books by Markeev [1] and Meyer, Hall and Offin [2] one finds the answer to the stability question in all these resonant cases except the 1 : −1 case considered here. At this equilibrium the Hamiltonian is indefinite and the linearized equations are weakly unstable. We give a fairly simple geometric proof of a general theorem which when applied to the Lagrange points L4 and L5 shows that these equilibrium points are Liapunov stable. In fact we show that the origin is encircled by invariant tori. The proof depends on a clear understanding of the local geometry, coordinates adapted to the geometry, and Moser’s invariant curve theorem [3]. The problem has been treated by several authors [4–8], but some of the arguments are in question. See the recent paper by Lerman and Markova [6] for a critical review of the literature and a detailed exhausting proof of this stability result. *

E-mail: [email protected] E-mail: [email protected] *** E-mail: [email protected] **

24

STABILITY OF A HAMILTONIAN SYSTEM IN A LIMITING CASE

25

The basic hypothesis on the Hamiltonian H is given in Section 2 along with the statement of the main theorem and its application to the restricted three body problem. Section 3 discusses the geometry of a neighborhood of the origin in the level set H = 0 which sets the stage for the construction of the cross section and the section map given in Section 4. Section 5 proves that the origin is stable in the level set H = 0 by showing that Moser’s invariant curve theorem applies to the constructed cross section map. The proof is finished in Section 6 by showing the argument of the previous section extends to small nearby level sets of H. In order not to disrupt the flow of the argument several digressions appear in the appendices. 2. STATEMENT OF THE THEOREM Consider a Hamiltonian 1 T z Sz + Hh (z) = H2 (z) + Hh (z), 2 and the corresponding equations of motion H(z) =

z˙ = Az + J∇Hh (z),

z ∈ R4

A = JS,

(2.1)

(2.2)

where H(z) is real analytic in a neighborhood of the origin which is a critical point, Hh (z) starts with cubic terms, S, A, J are 4 × 4 matrices, S symmetric, A Hamiltonian, J the usual matrix of Hamiltonian theory, and z˙ = dz/dt, t ∈ R. Our first assumption is that the Hamiltonian matrix A has eigenvalues ±iω, ω = 0, with multiplicity two and A is not diagonalizable. In this case the normal form for a quadratic Hamiltonian (linear Hamiltonian system) is: δ H2 = ω(x2 y1 − x1 y2 ) + (x21 + x22 ), 2 where δ = ±1. The linear system of equations is z˙ = Az, where ⎡ ⎤ ⎡ ⎤ ⎢ 0 ω 0 0⎥ ⎢ x1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ −ω 0 0 0 ⎥ ⎢ x2 ⎥ ⎥ ⎢ ⎥. ⎢ A=⎢ z = ⎢ ⎥, ⎥ ⎢ −δ 0 0 ω ⎥ ⎢ y1 ⎥ ⎣ ⎦ ⎣ ⎦

(2.3)

(2.4)

0 −δ −ω 0

y2

The four invariants usually associated with this Hamiltonian are: Γ1 = x2 y1 − x1 y2 ,

1 Γ2 = (x21 + x22 ), 2

1 Γ3 = (y12 + y22 ), 2

Γ4 = x1 y1 + x2 y2 .

Consider the nonlinear Hamiltonian system which has H2 as its quadratic part and has been normalized to Sokol’skii normal form through the fourth–order terms; i.e., consider 1 (2.5) H = ωΓ1 + δΓ2 + (aΓ21 + 2bΓ1 Γ3 + cΓ23 ) + · · · , 2 where the ellipsis stands for terms that are at least fifth-order. Use the uneven scaling first introduced by Meyer and Schmidt in [9]: x1 → ε2 x1 , x2 → ε2 x2 ,

(2.6)

y1 → εy1 , y2 → εy2 , which is symplectic with multiplier ε−3 ; so, the Hamiltonian becomes 1 H = ωΓ1 + ε(δΓ2 + cΓ23 ) + O(ε2 ). 2

(2.7)

Sokol’skii [7, 8] has shown that the origin is unstable if δc < 0 — also see [2]. Our main result is REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

26

MEYER et al.

Theorem 1. If δc > 0 the origin is stable for the system whose Hamiltonian is (2.5). In the planar circular restricted three-body problem when the mass ratio parameter μ is: √ 1 μR = (1 − 69/9) ≈ 0.0385209, 2 the exponents of the libration points L4 and L5 are a pair of pure imaginary numbers of multiplicity two. Schmidt [10] put the Hamiltonian of the restricted three-body problem at L4 and L5 into the normal form (2.5) with √ 59 2 , δ = 1, c= . (2.8) ω= 2 864 The value for c agrees with the independent calculations in Niedzielska [11] and Go´zdziewski and Maciejewski [12]. It agrees with the value computed in a different coordinate system by Deprit and Henrard [13]. It differs from the numeric value given by Markeev [1]. By these considerations and calculations we have the following. Corollary 1. The Lagrange equilibrium points L4 and L5 of the planar circular restricted threebody problem are Liapunov stable when the mass ratio parameter is equal to the critical Routh value μR . 3. THE LOCAL GEOMETRY Our problem is a small perturbation of the Hamiltonian Γ1 which the reader will recognize as the angular momentum when x ∈ R2 stands for the coordinates and y ∈ R2 stands for its conjugate momentum. So henceforth it will be denoted by Γ, i.e., Γ ≡ Γ1 = x2 y1 − x1 y2 .

(3.1)

Since this paper considers a perturbation of Γ it is important that we spend some time investigating the geometry of this Hamiltonian. By making the linear symplectic change of coordinates x1 =

√1 (u1 2

− v2 ), y1 =

√1 (u2 2

+ v1 ),

x2 =

√1 (u2 2

− v1 ), y2 =

√1 (u1 2

+ v2 ),

the Hamiltonian becomes 1 (−u21 + u22 − v12 + v22 ), (3.2) 2 thus manifesting that Hamiltonian (3.1) is equivalent to the quadratic part of a Hamiltonian system with semisimple 1 : −1 resonance. On the integral manifold where Γ = γ > 0 we have Γ ≡ Γ(u, v) =

u22 + v22 − 2γ = u21 + v12 = ρ2 .

√ So above each point P in the u2 v2 –plane outside the (blue) circle of radius 2γ there is a circle of radius ρ in the u1 v1 –plane and above each point p on the blue circle there is a point (the origin) in the u1 v1 –plane, see Fig. 1a. Thus above the (green) dashed ray in the u2 v2 –plane through p and P there lies the whole u1 v1 –plane. Letting the ray rotate all the way around in the u2 v2 –plane we get a solid torus, S 1 × R2 . When Γ = γ < 0 the picture is the same with the subscripts reversed. Thus, the energy surface Γ = γ = 0 is a three–dimensional hyperboloid that is homeomorphic to a solid torus. Now imagine the energy surface Γ = γ as γ → 0. The inner (blue) circle tends to a point and so the algebraic variety where Γ = 0 is not a manifold, but is homeomorphic to a solid torus with the center (blue) circle identified to a point, i.e., S 1 × R2 /(S 1 × (0, 0)), see Fig. 1b. REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

STABILITY OF A HAMILTONIAN SYSTEM IN A LIMITING CASE

27

Fig. 1. The integral manifold of Γ.

From (3.2) the equations of motion in the uv–space are: u˙ 1 = −v1 , u˙ 2 = v2 , v˙ 1 = u1 ,

v˙ 2 = −u2 ,

which represent the equations of two harmonic oscillators of period 2π. Thus after t increases by 2π both oscillators return to their initial value. So each orbit intersects the plane above the (green) dashed ray once and only once which we may take as a cross section to the flow on the integral surface Γ = γ. Another convenient cross section is introduced in the next section. 4. COMPUTING A CROSS SECTION MAP Let us compute a section map to the flow. We start by transforming Hamiltonian (2.7) using the change introduced in Section 3 to emphasize the underlying 1 : −1 resonance. The result is: H =

ω (−u21 − v12 + u22 + v22 ) 2



2  ε 8δ (u2 − v1 )2 + (u1 − v2 )2 + c (u2 + v1 )2 + (u1 + v2 )2 + O(ε2 ). + 32

(4.1)

Then introduce the standard action–angle coordinates: u1 = (2I˜1 )1/2 cos θ˜1 , v1 = (2I˜1 )1/2 sin θ˜1 , u2 = (2I˜2 )1/2 cos θ˜2 , v2 = (2I˜2 )1/2 sin θ˜2 , (note that du1 ∧ dv1 + du2 ∧ dv2 = dI˜1 ∧ dθ˜1 + dI˜2 ∧ dθ˜2 ) to get: H = ω(I˜2 − I˜1 ) ε + 4δ(I˜1 + I˜2 ) + c(I˜12 + 4I˜1 I˜2 + I˜22 ) − 2cI˜1 I˜2 cos(2(θ˜1 + θ˜2 )) 8 

1/2 1/2 − 2δ + c(I˜1 + I˜2 ) sin(θ˜1 + θ˜2 ) + O(ε2 ). + 4I˜1 I˜2 Next make the symplectic change: I˜1 = 12 (I1 + I2 ), I˜2 = 12 (−I1 + I2 ), θ˜1 = θ1 + θ2 , REGULAR AND CHAOTIC DYNAMICS

Vol. 17

θ˜2 = −θ1 + θ2 , No. 1

2012

(4.2)

28

MEYER et al.

arriving at the Hamiltonian: H = −ωI1 +

ε H1 + O(ε2 ), 16

(4.3)

where H1 = −cI12 + 8δI2 + 3cI22 + c(I12 − I22 ) cos 4θ2 + 4(cI2 − 2δ)(I22 − I12 )1/2 sin 2θ2 .

(4.4)

Note that I1 = −Γ1 and I2  0. The equations of motion are: ε ∂H1 I˙2 = + O(ε2 ), 16 ∂θ2 ε ∂H1 ε ∂H1 + O(ε2 ), θ˙2 = − + O(ε2 ). θ˙1 = ω − 16 ∂I1 16 ∂I2 I˙1 = O(ε2 ),

(4.5)

Compute the section map in the energy level H = −ωg where g is a parameter. In that level I1 = g + O(ε). Take the cross section to be θ1 = 0 and use I2 , θ2 as coordinates in that section. For ethic reasons we drop the subscript 2 in the following. The first return time is T = 2π/ω + O(ε) and so the section map is P : (I, θ) → (I ∗ , θ ∗ ) with ∂L + O(ν 2 ), ∂θ ∂L + O(ν 2 ), θ ∗=θ − ν ∂I

(4.6)

L = 8δI + 3cI 2 + c(g2 − I 2 ) cos 4θ + 4(cI − 2δ)(I 2 − g2 )1/2 sin 2θ.

(4.7)

I ∗=I + ν

where ν = επ/(8ω) and We will treat L like a generating function for P. Note that L is π–periodic in θ. Treating ν as small is equivalent to treating ε as small. The map P is a symplectic diffeomorphism (symplectomorphism) of a special form — see Appendix A. We have written this diffeomorphism in this form to illustrate its connection to the following Hamiltonian differential equation ∂L dI = , dν ∂θ

dθ ∂L =− . dν ∂I

(4.8)

For fixed ν the solutions of these equations are of the form (4.6) with possible different O(ν 2 ) terms. Since in general it is easier to change variables in a differential equation than in a diffeomorphism we shall exploit this form of the equations. 5. STABILITY WHEN H = 0 Lemma 1. When δc > 0 the origin is Liapunov stable in the Hamiltonian level set H = 0 for the system whose Hamiltonian is (2.5). Proof. Consider the flow on the energy level H = 0 by setting g = 0 so that L becomes L0 = 8δI(1 − sin 2θ) + cI 2 (3 − cos 4θ + 4 sin 2θ).

(5.1)

In this case I = 0 is a fixed point for the diffeomorphism P in (4.6) and an equilibrium point for Eqs. (4.8). As L0 in (5.1) can be rewritten as L0 = 16δI sin2 (θ − π/4) + 8cI 2 sin4 (θ + π/4)

(5.2)

we introduce coordinates q and p such that 16δI sin2 (θ − π/4) = Aq 2 ,

8cI 2 sin4 (θ + π/4) = Bp4 ,

REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

STABILITY OF A HAMILTONIAN SYSTEM IN A LIMITING CASE

29

where A = 8δ and B = c/4. That is, we define the symplectic change:

1 2 2 −1 p + q , θ = tan I = (q + p ), 2 p−q whose inverse is:

√

q = ± I cos θ − sin θ ,

√

p = ± I cos θ + sin θ .

Applying this change to L0 gives L0 = 8δq 2 + 2cp4 .

(5.3)

The terms inside O(ν 2 ) of (4.6) are analytic functions in q and p; indeed they are polynomials in the two coordinates. The plots in Fig. 2 are the level curves of L0 around the point q = p = 0 for δc positive and negative. The plot for δc positive looks like a nonlinear center and the plot for δc negative looks like a nonlinear saddle.

Fig. 2. Level curves of L0 : on the left for δc > 0, on the right δc < 0.

We concentrate on the case δc > 0 in order to prove the stability of the fixed point at the origin in R2 when L0 is (5.1). Scale q and p as follows: p q √ , p→ √ , q→ 32δ δc 4 δc which is a symplectic change with multiplier (128δ2 c)−1 so that L0 = q 2 + p 4 .

(5.4)

Introduce a type of polar coordinates (ρ, α) as was done in [14]. In particular define q = ρ2 η(α),

p = ρξ(α),

(5.5)

where ξ(α) is the sine lemniscate function and η = dξ/dα — see Appendix B. As ρ and α are not canonically conjugate we will change variables using the ideas in Appendix A, i.e., considering the associated differential equation. Start with dq = 4p3 , dν

dp = −2q, dν

REGULAR AND CHAOTIC DYNAMICS

Vol. 17

dξ = η, dα No. 1

2012

dη = −2ξ 3 . dα

30

MEYER et al.

Substituting change (5.5) into the equations for the derivatives of q and p above gives: dα dρ dα dρ = 4ρ3 ξ 3 , ξ + ρη = −2ρ2 η. 2ρ η − 2ρ2 ξ 3 dν dν dν dν Multiplying the first equation by η and the second by 2ρξ 3 and adding the resulting equations gives dρ/dν = 0. This implies dα/dν = −2ρ. Integrating these equations yields that the section map P : (ρ, α) → (ρ∗ , α∗ ) is: ρ∗ =ρ + O(ν 2 ), (5.6) α∗ =α

− 2νρ +

O(ν 2 ).

This map is periodic in α with period 2κ (since L was periodic in θ with period π). Besides, it is defined in {(ρ, α) | ρ  0, α ∈ [0, 2κ)} and is analytic in this domain. The map is a twist map since the coefficient of ρ in α∗ does not vanish. Since this map came from an area preserving mapping it satisfies the circle intersection hypothesis also so Moser’s invariant curve theorem [3] applies. In particular, there is a fixed ν0 > 0 such that when ν = ν0 for any β > 0 the map P has an invariant curve of the form ρ = ρ˜(α) = ρ˜(α + 2κ) > 0 with ρ˜(α)  β. (In fact there are infinitely many such invariant curves.) This shows that the fixed point at the origin of R2 for P is stable when ν = ν0 . Let ε0 correspond to ν0 , i.e., ν0 = ε0 π/(8ω). The stability for P in turn implies the origin of R4 is stable for the equations of motion (4.5) when ε = ε0 and H = 0 or g = 0. For fixed ε0 the scaling (2.6) is invertible. So all the transformations up until this point are invertible and thus the origin of R4 is stable for (2.5) in H = 0 and moreover it is encircled by invariant 2–tori that are reconstructed from the invariant curves of the section map given in (4.6). 6. COMPLETION OF THE PROOF OF THEOREM 1 When H = 0 Eqs. (2.5) have two families of elliptic periodic solutions, one for positive H and one for negative H, which are close to the origin for small ν. These are the Buchanan solutions [15] which appear as fixed points of the periodic map P given in (4.6). In Appendix C we include a proof of their existence which has been adapted from the proof of the Hamiltonian–Hopf bifurcation found in [2, 9]. We include the proof to emphasize that these solutions written in rectangular coordinates are analytic in all the parameters. That means that the map that shifts the origin of our coordinate system to the fixed point is analytic. Thus the period map will be analytic in rectangular coordinates or it has the d’Alembert character [2] in action–angle variables. Proof. For all g the section map P is of the form (4.6) where ∂L =−4c(I 2 − g2 ) sin 4θ + 8(2δ − cI)(I 2 − g2 )1/2 cos 2θ, ∂θ   ∂L =8δ + 2cI(3 − cos 4θ) + 4 c(I 2 − g2 )1/2 + I(cI − 2δ)(I 2 − g2 )−1/2 sin 2θ, ∂I from which one deduces that P has a fixed point, (I0 , θ0 ), of the form

(2|δ|)2/3 + |gc|2/3 |g|2/3 π + O(ν), θ0 = + O(ν), I0 = 4/3 1/3 4 2 (δc)

(6.1)

(6.2)

provided that δc > 0 and 0 < |g| < 2(δ/c). As g → 0, the point (I0 , θ0 ) tends to the origin of R2 . By Proposition 1 this fixed point is linearly stable and we show here that there are invariant curves encircling it at least for very small g. To that end let g = ν 3/2 d so L = L0 and I0 = O(ν) when |d|  1. This puts us into the case H = 0 and one repeats the proof of Lemma 1. Thence, for all |d|  1 and one fixed ν0 , the point (I0 , θ0 ) is encircled by invariant curves, thus it is a stable fixed point for the section map P. Note that the case g = 0 or H = 0 is included. Reconstructing the flow of Eqs. (4.5) and fixing ε = ε0 , the Buchanan periodic solutions are encircled by invariant 2–tori for ε0 small, thus they are orbitally stable periodic solutions for (2.5) in the energy level H = −ωg, with g small. As these periodic solutions surround the origin of R4 , this gives a full neighborhood of the origin filled with invariant 2–tori, thus this is a stable point for Hamiltonian (2.5). REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

STABILITY OF A HAMILTONIAN SYSTEM IN A LIMITING CASE

31

APPENDIX A. DIFFEOMORPHISMS AND DIFFERENTIAL EQUATIONS Consider the differential equation dz = f (z), (6.3) dν where f is analytic in an open set in Rm and ν ∈ R. Let z(z, ν) be the solution of (6.3) such that z(z, 0) = z. For each fixed ν the map z → z(z, ν) is a diffeomorphism and for small ν one has z(z, ν) = z + νf (z) + O(ν 2 ). We will say that Eq. (6.3) defines the diffeomorphism z → z ∗ where z ∗ = z + νf (z) + O(ν 2 ).

(6.4)

The fact that a differential equation can be used to generate a near identity diffeomorphism is the essence of the method of Lie transforms [16]. Let us change variables by z = Z(ζ), so that Eq. (6.3) becomes

−1 ∂Z dζ = (ζ) f (Z(ζ)) = f˜(ζ). (6.5) dν ∂ζ Conversely, let (6.4) be a given diffeomorphism and change variables by z = Z(ζ). Apply Z −1 to both sides of (6.4) to get:

ζ ∗ = Z −1 (z ∗ ) = Z −1 z + νf (z) + O(ν 2 )

−1 ∂Z −1 −1 (Z (z)) f (z) + O(ν 2 ) = Z (z) + ν ∂ζ

−1 ∂Z (ζ) f (Z(ζ)) + O(ν 2 ) =ζ + ν ∂ζ = ζ + ν f˜(ζ) + O(ν 2 ). We see that the differential equation and the coefficient of ν in the diffeomorphism both transform by the same rule under a change of variables, i.e.,

−1 ∂Z (ζ) f (Z(ζ)). (6.6) f (z) → f˜(ζ) = ∂ζ That is they transform as contravariant vectors. Of course the Hamiltonian character of f would be preserved if Z is symplectic. The reader should be warned not to carry this association too far. It is only valid to the first order in the parameter ν and so general theorems about differential equations do not automatically carry over to diffeomorphisms. APPENDIX B. THE SINE LEMNISCATE FUNCTION Elliptic functions have long been the domain of complex analyst, but a simple dynamical introduction will suffice for our goals. The sine lemniscate function, denoted by sin lemn α or sl α can be defined as the solution of the differential equation ξ  + 2ξ 3 = 0,

ξ  (0) = 1,

(6.7)

η 2 (α) + ξ 4 (α) ≡ 1 with η(α) = ξ  (α),

(6.8)

ξ(0) = 0,



where = d/dα. This equation has an integral given by

which implies ξ(α) is periodic — see Fig. 3a. Let κ be the least positive value such that ξ(κ) = 1. Solving (6.8) for η = dξ/dα and then separating variables one finds

 1 1 1 1 dτ √ , ≈ 1.311028777 . . . , = B κ= 4 4 2 1 − τ4 0 REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

32

MEYER et al.

Fig. 3. The sine lemn function: (a) its integral and (b) its graph.

where B is the classical Beta function. By symmetry arguments ξ(α) is odd and even about α = κ, i.e., ξ(κ + α) ≡ ξ(κ − α), and therefore ξ(α) is 4κ–periodic — see Fig. 3b. The traditional approach defines the inverse of sin lemn as an indefinite integral and therefore the sin lemn is the inverse of an elliptic integral – see [17] page 524. It is stated there that the idea of investigating this function occurred to C. F. Gauss on January 8, 1797 and it represents the first appearance of functions defined as the inverse of an integral. APPENDIX C. THE BUCHANAN PERIODIC SOLUTIONS In an interesting paper Buchanan [15] proved up to a computation of a higher–order term that there are two families of periodic solutions emanating from the libration point L4 (or from the point L5 ) even when μ = μR . This is particularly interesting, because the linearized equations have only one family. The computation of the higher–order term was completed by Deprit and Henrard [16], thus showing that Buchanan’s theorem did indeed apply to the restricted problem. Here we will reprove Buchanan’s theorem following the presentation of the Hamiltonian–Hopf theorem in [2, 9], which will be used in the proof that these solutions are orbitally stable thus completing the proof of Theorem 1. Proposition 1. There are two families of elliptic periodic solutions emanating from the origin for the system whose Hamiltonian is (2.5) if δc > 0. One family is parameterized by positive H and one by negative H. Sokol’skii has shown that there are no such periodic solutions when δc < 0. Proof. Introduce new symplectic complex coordinates by χ1 =

√1 (x1 2

+ ix2 ), χ2 =

√1 (x1 2

− ix2 ),

χ3 =

√1 (y1 2

− iy2 ), χ4 =

√1 (y1 2

+ iy2 ).

¯2 and χ3 = χ ¯4 . So Note that the reality conditions are χ1 = χ Γ1 = i(χ2 χ4 − χ1 χ3 ),

Γ 2 = χ1 χ2 ,

Γ 3 = χ3 χ4 ,

Γ 4 = χ1 χ3 + χ2 χ4 .

The equations of motion are: χ˙ 1 = −iωχ1 + εc(χ3 χ4 )χ4 , χ˙ 2 = iωχ2 + εc(χ3 χ4 )χ3 , (6.9) χ˙ 3 = iωχ3 − εδχ2 , χ˙ 4 = −iωχ4 − εδχ1 . REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

STABILITY OF A HAMILTONIAN SYSTEM IN A LIMITING CASE

33

Note that the O(ε2 ) terms have been dropped for the time being. Equations (6.9) are of the form ζ˙ = Cζ + εF (ζ), (6.10) where ζ is a 4–vector, F is analytic and F (0) = 0,

C = diag(−iω, iω, iω, −iω),

exp(CT ) = I4 ,

F (eCt ζ) = eCt F (ζ),

with T = 2π/ω. This last property is the characterization of the normal form when the matrix of the linear part is semisimple. Let τ be a parameter (period correction parameter or detuning); then ζ(t) = e(1−ετ )Ct υ, υ a constant vector, is a solution if and only if D(τ, υ) = τ Cυ + F (υ) = 0.

(6.11)

Thus if υ satisfies (6.11), then e(1−ετ )Ct υ is a periodic solution of (6.10) of period T /(1 − ετ ) = T (1 + ετ + · · · ). For Eqs. (6.9) with the O(ε2 ) terms omitted, one calculates ⎤ ⎡ 2χ + cs −iωτ χ 1 4⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ iωτ χ + cs2 χ ⎥ 2 3 ⎥ ⎢ ⎥ = 0, (6.12) D(τ, χ) = ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ iωτ χ3 − δχ2 ⎥ ⎥ ⎢ ⎦ ⎣ −iωτ χ4 − δχ1 where s2 = χ3 χ4 . Solving for χ1 from the last equation, substituting it into the first equation, and canceling the χ4 yields (ωτ )2 − δcs2 = 0.

(6.13)

A solution of (6.13) yields two families of periodic orbits, one for τ > 0 and one for τ < 0. Solve (6.13) for s2 to get s2 = (ωτ )2 /(δc). Take χ3 = a1 + ia2 and χ4 = a1 − ia2 with s2 = χ3 χ4 , then solve for χ1 and χ2 by the last two equations in (6.12). Fixing s2 determines two circles of periodic solutions, or two periodic orbits.

Fig. 4. τ versus s.

Recall that δc > 0. The graph of (6.13) represents τ versus s and is two lines through the origin, but only the part where s  0 is of interest — see Fig. 4. The parameter τ is the correction to the period. By the paragraph above, a fixed solution of (6.13), s = 0, fixes the length of z3 and so fixes the coordinates z1 , z2 , z3 , z4 up to a circle. Thus a point in the τ s–plane, s = 0, on the graph of (6.13) corresponds to a periodic orbit of (6.9) with period T /(1 − ετ ) whereas s = 0 corresponds to the origin. The invariants on these periodic solutions are: 2(ωτ )3 2ωτ s2 (ωτ )4 (ωτ )2 =− , Γ2 = (ωτ s)2 = , Γ3 = s 2 = , Γ4 = 0. δ c δc δc Note that by the above the two families are also parameterized by Γ1 and hence by H for small ε. Γ1 = −

REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

34

MEYER et al.

These conclusions remain valid when the O(ε2 ) terms in (6.9) are present. If the O(ε2 ) terms are included, (6.9) is of the form ζ˙ = Cζ + f (ζ) + O(ε2 ). Let φ(t, υ, ε) be the general solution of this equation with φ(0, υ, ε) = υ. Let Φ(υ, τ, ε) be this solution after a time T (1 + ετ ); i.e.,

Φ(υ, τ, ε) = φ T (1 + ετ ), υ, ε = υ + εD(τ, υ) + O(ε2 )

= υ + ε τ Cυ + F (υ) + O(ε2 ). A periodic solution corresponds to a solution of Φ(υ, τ, ε) = υ, and so the equation to be solved is:



D(υ, τ, ε) = Φ(υ, τ, ε) − υ /ε = D(τ, υ) + O(ε) = 0.

(6.14)

Equations (6.14) are dependent because Eqs. (6.9) admit H as an integral. Because H(υ + εD) = H(υ), it follows by the mean value theorem that ∇H(υ ∗ )D = 0, where υ ∗ is a point between υ and υ + εD. Because ∇H(υ ∗ ) = (−iωχ3 , . . .) + · · · , if D2 = D3 = D4 = 0, then D1 = 0, except maybe when χ3 = 0. Thus only the last three equations in (6.14) need to be solved because the solutions sought have χ3 = 0. From the last two equations in (6.14), one solves by the implicit function theorem for χ1 and χ2 to get χ1 = −iωτ δχ4 + · · · , χ2 = iωτ δχ3 + · · · . Substitute these solutions into the second equation to get: K(χ3 , χ4 , τ, ε) = (ω 2 τ 2 − δcs2 )(−δχ3 ) + εk(χ3 , χ4 , τ, ε) = 0.

(6.15)

Because the origin is always an equilibrium point, K and k vanish when χ3 and χ4 are zero. Let χ3 = seiα , χ4 = se−iα , and divide (6.15) by −δs to get: ˜ α, τ, ε) = 0. ˜ α, τ, ε) = (ω 2 τ 2 − δcs2 )eiα + εk(s, K(r, 2 2 2 Because δc > 0, this equation can be  solved for s to get s = (ωτ ) /(δc) + · · · for all α, all τ , |τ | < τ0 , and all ε, |ε| < ε0 . So s = ± (ωτ )2 /(δc) + · · · are real solutions.

The characteristic multipliers are the eigenvalues of ∂Φ/∂υ evaluated along the solution. Compute ⎤ ⎡ 2 2 −iωτ 0 cχ4 2cs ⎥ ⎢ ⎥ ⎢ ⎢ 2 cχ2 ⎥ ⎥ ⎢ 0 iωτ 2cs 3 ⎥ ∂D ⎢ =⎢ ⎥ ⎥ ⎢ ∂υ ⎢ 0 −δ iωτ 0 ⎥ ⎥ ⎢ ⎦ ⎣ −δ 0 0 −iωτ  whose eigenvalues evaluated along (6.13) are 0, 0, and ±i 3δc(ωτ )2 and so the multipliers are:   1, 1, 1 + εi 3δc(ωτ )2 + O(ε2 ), 1 − εi 3δc(ωτ )2 + O(ε2 ). Thus the Buchanan periodic solutions are elliptic. Corollary 2. There are two families of elliptic periodic solutions emanating from the Lagrange equilibrium points L4 and L5 of the planar circular restricted three-body problem when the mass ratio parameter is equal to the critical Routh value μR . Proof. Apply Proposition 1 with the specific values of ω, δ and c given in (2.8). REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012

STABILITY OF A HAMILTONIAN SYSTEM IN A LIMITING CASE

35

ACKNOWLEDGMENTS We acknowledge the comments of Professor Dieter Schmidt. The authors are partially supported by Project MTM 2008-03818 of the Ministry of Science and Innovation of Spain and by a grant from the Charles Phelps Taft Foundation. REFERENCES 1. Markeev, A. P., Libration Points in Celestial Mechanics and Cosmodynamics, Moscow: Nauka, 1978 (in Russian). 2. Meyer, K. R., Hall, G. R., and Offin, D., Introduction to Hamiltonian Dynamical Systems and the N -Body Problem, 2nd ed., New York: Springer, 2009. 3. Moser, J., On Invariant Curves of Area-Preserving Mappings of an Annulus, Nachr. Akad. Wiss. G¨ ottingen, Math.-Phys. Kl. II, 1962, vol. 6, pp. 1–20. 4. Dullin, H. R. and Ivanov, A. V., Vanishing Twist in the Hamiltonian Hopf Bifurcation, Phys. D, 2005, vol. 201, nos. 1–2, pp. 27–44. 5. Kovalev, A. M. and Chudnenko, A. M., On the Stability of the Equilibrium Position of a Two-Dimensional Hamiltonian System in the Case of Equal Frequencies, Dokl. Akad. Nauk Ukrain. SSR, 1977, vol. 11, pp. 1011–1014 (in Russian). 6. Lerman, L. M. and Markova, A. P., On Stability at the Hamiltonian Hopf Bifurcation, Regul. Chaotic Dyn., 2009, vol. 14, pp. 148–162. 7. Sokol’skii, A. G., On Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom under First-Order Resonance, Prikl. Mat. Mekh., 1977, vol. 41, no. 1, pp. 24–33 [J. Appl. Math. Mech., 1977, vol. 41, no. 1, pp. 20–28]. 8. Sokol’ski˘i, A. G., Proof of the Stability of Lagrangian Solutions for a Critical Mass Ratio, Sov. Astron. Lett., 1978, vol. 4, pp. 79–81. 9. Meyer, K. R. and Schmidt, D. S., Periodic Orbits near L4 for Mass Ratios near the Critical Mass Ratio of Routh, Celestial Mech., 1971, vol. 4, pp. 99–109. 10. Schmidt, D. S., Transformation to Versal Normal Form, Computer Aided Proofs in Analysis, K. R. Meyer and D. S. Schmidt (Eds.), IMA Series, vol. 28, New York: Springer, 1990. 11. Niedzielska, Z., Nonlinear Stability of the Libration Points in the Photogravitational Restricted Three Body Problem, Celestial Mech. Dynam. Astronom., 1994, vol. 58, pp. 203–213. 12. Go´zdziewski, K. and Maciejewski, A. J., Nonlinear Stability of the Lagrangian Libration Points in the Chermnykh Problem, Celestial Mech. Dynam. Astronom., 1998, vol. 70, no. 1, pp. 41–58. 13. Deprit, A. and Henrard, J., A Manifold of Periodic Orbits, Adv. Astron. Astrophys., 1968, vol. 6, pp. 1– 124. 14. Meyer, K. R., Generic Stability Properties of Periodic Points, Trans. Amer. Math. Soc., 1971, vol. 154, pp. 273–277. 15. Buchanan, D., Trojan Satellites (Limiting Case), Trans. Royal Soc. Canada, 1941, vol. 35, pp. 9–25. 16. Deprit, A., Canonical Transformations Depending on a Small Parameter, Celestial Mech., 1969/70, vol. 1, pp. 12–30. 17. Whittaker, E. T. and Watson, G. N., A Course of Modern Analysis, 4th ed., New York: Cambridge Univ. Press, 1996.

REGULAR AND CHAOTIC DYNAMICS

Vol. 17

No. 1

2012