On Stability of the Collinear Libration Point L1 in the Planar Restricted Circular Photogravitational Three-Body Problem Текст научной статьи по специальности «Физика»

Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 4, pp. 543-562. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221202

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 34D20, 37J40, 70K30, 70K45, 37N05

On Stability of the Collinear Libration Point Lx in the Planar Restricted Circular Photogravitational

Three-Body Problem

B.S.Bardin, A. N. Avdyushkin

The stability of the collinear libration point L1 in the photogravitational three-body problem is investigated. This problem is concerned with the motion of a body of infinitely small mass which experiences gravitational forces and repulsive forces of radiation pressure coming from two massive bodies. It is assumed that the massive bodies move in circular orbits and that the body of small mass is located in the plane of their motion. Using methods of normal forms and KAM theory, a rigorous analysis of the Lyapunov stability of the collinear libration point lying on the segment connecting the massive bodies is performed. Conclusions on the stability are drawn both for the nonresonant case and for the case of resonances through order four.

Keywords: collinear libration point, photogravitational three-body problem, normal forms, KAM theory, Lyapunov stability, resonances

1. Introduction

In the study of the motion of celestial bodies and spacecrafts, it is often necessary to take into account not only gravitational forces, but also forces of radiation pressure. In this case, the so-called restricted photogravitational three-body problem provides a good mathematical model describing the dynamics of space objects. This problem is concerned with the motion of a body of small mass, which experiences both gravitational forces and repulsive forces of radiation pressure coming from two radiating bodies [1]. A body of small mass does not affect the motion of massive

Received October 22, 2022 Accepted December 02, 2022

This research was supported by the Russian Foundation for Basic Research under the scientific projects No. 20-31-90064 and 20-01-00637.

Boris S. Bardin bsbardin@yandex.ru

Andrey N. Avdyushkin avdyushkin.a.n@ya.ru

Moscow Aviation Institute Volokolamskoye sh. 4, Moscow, 125080 Russia

radiating bodies which move in given Keplerian orbits. If it is additionally assumed that all three bodies move in the same plane, then this problem is called the planar circular restricted photogravitational three-body problem.

As in the classical three-body problem, the equations of motion of the photogravitational problem admit remarkable particular solutions [1] which describe the motions of bodies such that the body of small mass lies on a straight line connecting the radiating bodies. By analogy with the classical problem, such motions are called collinear (or rectilinear) libration points. They correspond to equilibrium points in a coordinate system rotating with the radiating bodies. The problem of the existence of collinear libration points was studied in detail in [2-4].

It is well known that in the classical three-body problem the collinear libration points are Lyapunov unstable (see, e.g., [5]). In [2, 3], it was shown for the first time that the influence of the repulsive forces of radiation pressure can lead to the linear stability of the collinear libration point L1, which is located on the segment between the attracting and radiating bodies. To date, there has been a large body of research undertaken to investigate the stability of the collinear libration points of the restricted photogravitational three-body problem. In [4, 6-9], an analysis was made of the stability of the collinear libration point L1, both in the planar circular and the elliptic restricted photogravitational three-body problem. For all possible parameter values, a linear analysis was carried out and diagrams of linear stability were constructed. For a number of special cases of the planar circular problem, a complete and rigorous analysis of Lyapunov stability [10-13] was carried out. In particular, in [10, 11] the problem of the Lyapunov stability of L1 in the presence of third- and fourth-order resonances was considered. In [12], the nonresonant and resonant cases were investigated under the condition that the absolute value of the repulsive forces of radiation pressure does not exceed that of the gravitational forces. In [13], under the same assumptions, rigorous conclusions on the Lyapunov stability of L1 were obtained for the cases of first- and second-order resonances corresponding to the boundaries of the linear stability of L1. In [14], a nonlinear stability analysis was carried out for a special case of the planar elliptic problem. It was assumed that the massive bodies have equal masses, act on the small body with equal repulsive forces of radiation pressure and move in weakly elliptic orbits. Conditions for the formal stability and instability of L1 were obtained.

The purpose of this paper is to obtain rigorous conclusions on the stability of L1 in the Lyapunov sense in the general case, when no restrictions are imposed on the parameter values.

2. Formulation of the problem and equations of motion

To describe the motion of a body of small mass P, we introduce a rotating coordinate system Oxyz with origin at the center of mass O of the system of two attracting and radiating bodies, P1 and P2, moving in circular orbits (see Fig. 1). Let the axis Ox be directed along a straight line passing through these bodies, let the axis Oz be perpendicular to the plane of their motion, and suppose that the axis Oy complements the coordinate system to the right triple.

Next, we assume that the body P moves in the plane of the orbits of the bodies P1 and P2, i.e., its position in the moving coordinate system Oxyz is completely specified by two coordinates x and y. We transform to dimensionless coordinates £ and n using the formulae

x = y = rn, r = const, (2.1)

where r = const is the distance between P1 and P2.

Pi

О

Fig. 1. Coordinate system

The equations of motion of the body P can be written in the following canonical form:

_ dH drj _ dH dp? _ dH dpv _ dH dt dp^ dt dp^ dt d£ ' dt dn '

with the Hamiltonian [10, 11]

(2.2)

where ri is the distance between the bodies P and P\ (i = 1, 2)

ri = V(t + VÏ2 + 'n2, r2 = y/tt + ll- l)2+ r?2,

m0

ß

m1 + m2

Q1 and Q2 denote the so-called coefficients of mass reduction [1], which characterize the forces of radiation pressure. The coefficients Qi (i = 1, 2) can take values from the region (-œ; 1]. If Qi < 0, then the absolute value of the repulsive forces exceeds that of the gravitational forces. In the limiting case Q1 = Q2 = 1, the forces of radiation pressure are zero.

To collinear libration points there corresponds the following steady-state solutions of the equation of motion (2.2)

С = С*, П = 0, Pç = 0, pv = С*, (2.3)

where is a real root of the equation

L -

Qi(i-ju)

(£* + ß)\L + ß\ (£* + ß~ + ß~ i|

Q2ß

0.

(2.4)

Next, we will consider only solutions (2.3) for which ^ £ (—ß; 1 — ß). In this case, Eq. (2.4) takes the form

Q ^1-fj) дФ _ ^ {L + ßY ' ( j

and solution (2.3) describes the libration point Ll lying inside the segment connecting the bodies P1 and P2 (Fig. 2).

pi

1

ri

Li

° L €

Fig. 2. The collinear point Lx

Next, we examine the problem of the Lyapunov stability of the libration point L1 for all admissible parameter values of the problem.

3. On the existence and bifurcation of the libration point Lx

In the classical three-body problem where Q1 = Q2 = 1, Eq. (2.5) has a unique solution for any admissible values of the parameter j on the interval (—j; 1 — j). By contrast, in the photogravitational three-body problem, depending on the values of the parameters Q1, Q2 and j, Eq. (2.5) can have, on the above-mentioned interval, one, two or three solutions, or may have no solutions at all. A qualitative analysis of the behavior of the function F1(£) on the interval (—j; 1 — j) shows that, at a fixed value of the parameter j, the number of roots of Eq. (2.5) changes either under a change of the sign of the parameters Q1, Q2 or at a bifurcation of solutions which occurs under the conditions

dF

F(U = 0, = (3.1)

where F(£) denotes the left-hand side of Eq. (2.5), i.e.,

Fin * _ Qi^-ri ,_<hE__(3 2)

(C + aO2 + )2" ( }

Expressing Q1 and Q2 from Eqs. (3.1), we have

_ (3e,+A*-i)(e*+A*)3 n _ (3e,+At)(e*+^-i)3

Ql 2(Wl) ' Q2 2^ • {d'd)

Relations (3.3) in parametric form define the surface equation in the three-dimensional space of the parameters j, Q1 and Q2. In what follows, we will call this surface a bifurcation surface, and the values of the parameters j, Q1 and Q2 on this surface, bifurcation values.

At a fixed value of j, the formulae (3.3) are equations of a curve in the plane Q1, Q2 which are written in parametric form, where the parameter takes values from the interval (—j; 1 — j). In what follows, we will call this curve a bifurcation curve. This curve is shown in Fig. 3 for j = 0.45.

The bifurcation curve and the coordinate axes divide the set of values of the parameters Q1 and Q2 into seven regions (see Fig. 3). Inside each of these regions the number of roots of Eq. (2.5) remains unchanged. In particular, in regions I and IIIb Eq. (2.5) has one root, in regions IIa and IVa two roots, in region IIIa three roots, and in regions IIb and IVb Eq. (2.5) has no roots. As the value of the parameter j changes, a deformation of these regions occurs, but qualitatively their configuration does not change. The latter implies that a similar situation takes place in the entire three-dimensional space of the parameters j, Q1 and Q2: the bifurcation surface and the coordinate planes Q1 = 0, Q2 = 0 divide the entire set of parameter values into seven regions such that the number of roots of Eq. (2.5) in each of them remains unchanged.

To solve the problem of the stability of the libration point, it is necessary to investigate the motion in its small neighborhood. To this end we introduce local coordinates (perturbations) qi, pi (i = 1, 2) using the formulae

£ = £* + Q1, n = Q2, Pi = P1, Pv = £* + P2.

Then the motion of the small mass in a neighborhood of the libration point (the perturbed motion) is described by a canonical system for which the series expansion of the Hamiltonian function in powers of qi and pi has the form

H = H2 + H3 + H4 + ... , (3.4) RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 543 562_

Ha

q2

0.80.6-

Ilb

0.4

0.2

-0.2

Illb

-0.2-

0.2 0.4 0.6 0.8 Q1 1 IVb

0

Ilia IVa

Fig. 3. Bifurcation curve and existence regions of solutions to Eq. (2.5) for ¡i = 0.45

where

= ö (Pi +^2) +P1Q2-P2Q1 ~ OQi +

2

H3 = bqf ~

3c

H4 = ~cq1 + 3cgl(?2 - — g2\

(3.5)

(3.6)

(3.7)

The constant term in the Hamiltonian has been omitted, and the coefficients of the forms (3.5)-(3.7) calculated for ^ G (—¡; 1 — ¡) have the form

Qi(1 — ¡) Q2i

& + ß)3 & + ß - 1)3:

Q2ß

& + ß)4 & + ß - 1)4:

-ß) _ Q2ß (£*+ß)5 (£* + ß-i)5'

(3.8)

(3.9) (3.10)

In the variables qi and pi the problem of the stability of the libration point reduces to that of the stability of the equilibrium point qi = pi = 0 (i = 1, 2) of a canonical system with the Hamiltonian (3.4).

c

4. Linear stability analysis

We start the investigation of stability with linear stability analysis. To this end we consider a linear canonical system with the Hamiltonian (3.5). The characteristic equation of this system has the form

A4 - aA2 + 2A2 - 2a2 + a + 1 = 0.

(4.1)

When a £ [—0] U [§; l], Eq. (4.1) has roots with a nonzero real part. In this case, the equilibrium point of the linear system is unstable. By virtue of the Lyapunov theorem on linear stability [15], this implies instability of the equilibrium point in a complete nonlinear system with the Hamiltonian (3.4) as well. The latter implies Lyapunov instability of the libration point L1 corresponding to the root ^ of Eq. (2.5).

If a G (—0) U (|; l), then Eq. (4.1) has two pairs of purely imaginary roots: ±-iw1, ±iu)2, where

Wi

w2

1 = ( 1 - -a, + -\/9a2 - 8a

= 1 - -a - -\/9a2 -8a

1/2

1/2

(4.2)

2

2

In this case, the equilibrium point is linearly stable [9]. The condition a G (—0) U (|; l) defines two regions in the three-dimensional space of the parameters i, Q1 and Q2. The boundaries of these regions are given by the equations a = — a = 0 and a = §, a = 1, where a is defined by the expression (3.8). Each of these equations, along with Eq. (2.5), can be uniquely solved for Q1 and Q2. As a result, on the boundaries of the regions of linear stability we have the following dependences of Q1 and Q2 on i and ^:

for a = — -2

for a = 0:

f 8 for a = - :

9

for a = 1:

Q1 Q1 Q1 =

(3{*+Ai-!)(£*+A*)3 _ (3eL+At)(C*+At-l)3

j V2 ~

2(1 - ¡ ) 1 — I

(^ — 8i + 8)(& + i)3

9(1 — i)

Q1 = (£* +1)3,

Q2 =

, Q2

2i

1

9^

Q2 = —(l+1 — 1)3

(4.3)

(4.4)

(4.5)

(4.6)

Relations (4.3)-(4.6) define the equations of the boundaries of the stability regions in parametric form, where the parameters i and take values from the intervals (0; 1) and (—1 — ¡), respectively. Note that, the boundary of the stability region given by the equation a = — ^ coincides with the bifurcation surface (3.3).

Figure 4 shows a section formed by the intersection of the stability regions in the parameter space of the problem with the plane i = 0.45. In this section, the linear stability regions are shown in gray. These regions adjoin the points (0; 1) and (1; 0) of the plane of the parameters Q1 and Q2. The upper region corresponds to values of the coefficient a from the interval (§; l), and the lower region, to the values of a from the interval (—0). As the value of the parameter ¿t approaches zero, these regions stretch down along the negative direction of the axis Q2, and as the value of i approaches unity, they stretch to the left along the negative direction of the axis Q1 (see Fig. 5).

c, , .,., . c n 1 Stability regions for a = 0.9

Stability regions for a = 0.1 jo/-

Fig. 5. Stability regions near boundary values of a

The linear stability of the equilibrium point does not imply its stability in the complete system with the Hamiltonian (3.4), and so, to draw conclusions on the Lyapunov stability of the

libration point L1, it is necessary to carry out a nonlinear analysis, taking into account terms of degree higher than two in the expansion of the Hamiltonian (3.4).

5. Stability analysis in the nonresonant case

In what follows it is assumed that a e [—0] U [§; l], i. e., we will consider the case where the parameter values of the problem belong to linear stability regions or to their boundaries. Rigorous conclusions on the Lyapunov stability of the libration point L1 in this case can be drawn on the basis of sufficient conditions for the stability of an autonomous Hamiltonian system with two degrees of freedom, which were obtained by methods of KAM theory [16-19]. To determine whether these sufficient conditions hold, we first need to normalize the Hamiltonian (3.4), i.e., to make a canonical change of variables that reduces the Hamiltonian to the so-called normal form which is the most convenient for stability analysis.

The normal form in the nonresonant case and the normal form in the resonant cases are different. Therefore, the resonant cases should be considered separately from the nonresonant case. Resonances of order one (u2 = 0) and of order two (w1 = u2) take place on the boundaries of the linear stability regions. They will be considered in Sections 7 and 8. Other resonances occur at internal points of the linear stability regions. A key role in the stability analysis is played by third- and fourth-order resonances. In our problem, the third-order resonance (w1 = 2u2)

occurs when a = a

where a3 =

41—5yT45 108 '

a+ =

41+5^/145 108 '

and the fourth-order resonance

_ 68-60V5 „+ _ 68+60V5

(u1 = 3u2) takes place when a - Q>4 , where Q^ - ggg , - ggg

In this section we assume that the system with the Hamiltonian function (3.4) has no resonances up to order four inclusive, i.e., the system is linearly stable and the relations a = a± and a = a± are satisfied.

The quadratic part of the Hamiltonian can be normalized using the linear canonical change of variables qi, pi ^ xi, yi made by a method described in [19, 20]. This change of variables has the form

q1 = a11x1 + a12x2 y Pi = b21 Vl + b22V2y

Q2 = biiVl + bi2V2y p2 = a21x1 + a22x2y

(5.1)

where

Ki (öj uf — a + 1 j

— 2au2 + a2 + 2u2 + 2a — 3

1/2

bu = 2SiKi, b2i = 5iKi (u2 - a + l), ¿1 = 1, 62 = -1, i = 1, 2.

In the variables xi, yi (i = 1, 2) the quadratic part of the Hamiltonian function takes the following form:

1 ,0 o^ 1

H2 = -wl (x\ + y\) - -u)2 (:x\ + yl).

2

(5.2)

After normalizing the quadratic part of the Hamiltonian, one can make a canonical near-identity change of variables xi, Vi ^ ui, vi (i = 1, 2), which reduces the Hamiltonian of the problem to normal form up to terms of some finite degree.

The generating function of such a change of variables has the following form:

5 = XiVi + 5(3)(xiyVi) (i = 1, 2),

(5.3)

where S(3"l(xi, vi) is a convergent power series that starts with terms of the third or higher degree. To calculate the coefficients of the generating function (5.3) and the coefficients of the normal form of the Hamiltonian, one can use the classical Birkhoff method [21] or the Deprit-Hori method [19, 22].

In the nonresonant case, the Hamiltonian normalized to terms of order four has the form

H = W1f1 — W2T2 + C20rf + Cn+ C02r2 + O ((^ + ^)5/2) , (5.4)

where the canonical variables ri, <pi (i = 1, 2) have been introduced by the formulae ui = = V^sin^, Vi = y/2Fi cos Ifii.

If A = c02w2 + c11w1w2 + c20w2 = 0, then, according to the Arnold-Moser theorem [16-18], the equilibrium point of the system with the Hamiltonian (5.4) is Lyapunov stable.

Using the Deprit-Hori method, the coefficients c02, c11, c20 were obtained and the following explicit expression for A was found:

35

A (a, b, c) - - 16a^9a _ ^ £54a2 _ 41q _ ^ ^ _ + , (5.5)

where

5 = 11664a7 c — 7776a6 b2 — 14 688a6 c + 6336a5 b2 — 8856a5 c+

+ 12 110a4 b2 + 17196a4 c — 19 009a3 b2 — 6483a3 c + 16 439a2 b2 —

— 3189a2 c — 5256ab2 + 3384ac — 2520b2 + 648c.

Thus, for the parameter values satisfying the inequalities a = a±, a = a± and A = 0, the libration point L1 is Lyapunov stable.

When A = 0, we have the case of degeneracy where the problem of stability cannot be solved by taking into account terms of order not higher than four in the expansion of the Hamil-tonian (3.4). To investigate stability in the case of degeneracy, it is necessary to take into account terms of at least order six. The equation A = 0 defines the two-dimensional surfaces of degeneracy in the linear stability regions of the three-dimensional parameter space of the problem. Figure 6 shows degeneracy curves for Qi ^ 0.4 (i = 1, 2) obtained by the intersection of the degeneracy surfaces with the plane i = 0.45.

The problem of the stability of the libration point L1 for parameter values corresponding to degeneracy surfaces is not investigated in this paper.

6. Stability analysis in the case of third- and fourth-order resonances

In the resonant cases, the normal form of the Hamiltonian contains additional resonant terms. In this section, we consider the cases of third- and fourth-order resonances. In the three-dimensional space of the parameters i, Q1 and Q2, to these resonances there correspond two-dimensional resonant surfaces given by the equations a = a±, a = a±, respectively, where a is defined by the formula (3.8).

Solving these equations together with Eq. (2.5) for Q1 and Q2, we have

Q i =--:-, Q2 =--> (6-1)

1 —i i

-0.1J

Fig. 6. Degeneracy curve (for j = 0.45)

where the quantity ar in (6.1) takes one of the resonant values, a = a± or a = a±, which correspond to a third- or fourth-order resonance, respectively.

Relations (6.1) are parametric equations of resonant surfaces. Here the parameters are i G G (0; 1) and ^ G (—1 — ¡). The latter parameter, ^, should be interpreted as a dimensionless coordinate of the libration point L1 (the root of Eq. (2.5)).

Resonant surfaces lie inside the linear stability regions. Figure 7 shows resonant curves for Qi ^ 0.4 (i = 1, 2) obtained by the intersection of resonant surfaces with the plane i = 0.45.

We first consider the third-order resonance u1 = 2u2, i.e., we set a = a±. In this case, the normalizing change of canonical variables reduces the Hamiltonian function to the following normal form [19]:

H = 2w^! - w2r2 -

v12'2

r2s/r\sm(ip1 +2ip2) + 0((r1 +r2)2).

(6.2)

Let k+2 denote the value of the coefficient k12 that corresponds to the case a = a+, and let k-2 denote the value of the coefficient k12 that corresponds to the case a = a-. The quantities k+~2 and k-2 obtained using the Deprit-Hori method have the form

k+2 =

k-2 = -

81 v/6 (61 /L45 - 731) (105 -2 (8360 - 640/L45) (475 -81 v/6 (61 v/l45 + 731) (105 +

2 (8360 + 640 /L45) (475 +

If k12 = 0, then, according to Markeev's theorem [19], the equilibrium point of the system with the Hamiltonian (6.2) is unstable, which implies instability of the collinear point L1. The special case k12 = 0 is possible only when b = 0. In this case, the normal form of the Hamiltonian contains no resonant part up to fourth-order terms inclusive. When k12 = 0, the normalized Hamiltonian has the form (5.4). Therefore, to solve the problem of stability, one can apply the above-mentioned Arnold-Moser theorem by virtue of which stability takes place if A = 0. Substituting a = a± and b = 0 into (5.5), we obtain

for a = a+, A

for a = a- , A

41 +

108

41 -

108

0, с ) = g^ i 617\Zl45 + 10 207) c;

0, с I = (617\/l45 — 10 207) c. 640

(6.3)

A special case takes place on the curve lying on the resonant surface in the three-dimensional space of the parameters i, Q1 and Q2. This curve is given by the equations a = a± and b = 0, where a and b are defined by the formulae (3.8) and (3.9). The equations of this curve can also be represented in the following parametric form:

^ _ + Q'-r ~ q _ _(a-r + £*)_ Q _ _(ar ~ __^g ^

2ar 8a2 (2a^++ ar ^^) 8a2((^ ++ ar 2ar)

where the parameter is the dimensionless coordinate of the libration point L1 and the quantity ar takes one of two values, a- or a+, which correspond to a third-order resonance.

Eliminating the parameter from (6.4), we can write the equations of this curve in the following simplest form:

_ (ar - ¿¿)4a..r _ (qr + fj, - 1 )4a..r

~ (2ar - 1)4(1 - ») ' ^ ~ (2ar - 1)V ' 1 j

We also note that in the formulae (6.5) the parameter i takes values from the interval (0; 1) when ar — a^ and from the interval (1 — a+; a+) when ar — a+ • This is easy to show, considering that G (—i; 1 — i) and taking into account the equation

a (2n - 1)

= 1 — 2ar ' (6"6)

which follows from the first of Eqs. (6.4).

Substituting (6.5) and (6.6) into (3.10), we obtain an expression for the coefficient c on the curve given by Eqs. (6.4)

c=7-(1 ~ 2a[}2ar ,. (6.7)

(ar +1 — 1)(ar — i)

Obviously, the numerator of the expression (6.7) does not vanish. Moreover, since a- < 0 and the parameter i takes, for ar — a+ , values from the interval (1 — a+; a+), it follows that the denominator of the expression (6.7) does not vanish either. Thus, the coefficient c given by the formula (6.7) is defined for all admissible values of the parameter i and does not vanish. Therefore, by virtue of the formulae (6.3), the quantity A is nonzero for a = a± and b = 0. Consequently, in this special case the collinear libration point L1 is Lyapunov stable.

We now consider the fourth-order resonance u1 = 3u2, i.e., we set a = a±. In the case of the fourth-order resonance, the normal form of the Hamiltonian has the form [19]

H = 3u}2r1 -U)2r2 + c20r2 + c11r1r2 + cmr22 +£r2V/r^sin(<£1 + 3(£2) + 0 (ir\ + r2)5/2). (6.8)

The coefficients of the normal form of the Hamiltonian (6.8) were calculated using the Deprit-Hori method. Let B± and C± denote the values of the coefficients B and C which correspond to the values a = a±:

, 10372745 ±2818766y/5 ( l9 / /- \ \

C* =---:-— ( 18833801632696 =F (305789811600V 5 ± 1132825190175 c ,

36368720394160000 V Tl J J'

, (47 ± 20 v/5)1/2 (36929 v/5=F 151550) // r- \ 9

J± - ^--i—----11390400V5 ± 19052675 c =F 4635975362

618130040000 VV J ^

B± =

618130040000

(6.9)

Since the coefficients b and c in (6.9) are calculated for values of parameters j, Q1, Q2 corresponding to the fourth order resonance, then the relations (6.1) take place. Thus, by combining formulas (3.9), (3.10), (6.1) and (6.9) the coefficients of normal form (6.8) can be written in terms of parameters j and Actually, we have the following formulas for C± and B ±

560 000({ + j)2(Ç + J - 1)2'

B ± =

F

±

(6.10)

40 000(e + j)2(C + j - 1)2' RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2022, 18(4), 543 562

where

G± = (l99 758 295 =F 88 566 074^) + ( (l64 139 670 =F 84 824 564^) ß ± 42 412 282y/b-- 82 069 835) £ + (265 190 980 ± 85 485 724 V^) - 1) + 11 012 620 =F 4 348 844 V^,

F ± =

794 050 + 354 *

+ ( ( 532 900 + 241'

+ i 193 500 ± 1908 v/5 J fJ,(fJ, - 1) + 20 500 =F 8548^5

ß ± 120 i

47

- 266 45m

(6-11)

According to Markeev's theorem [19], the canonical system with the Hamiltonian (6.8) is

stable in the sense of Lyapunov if the inequality |3\/3I3| < |C|, where C = c20 + 3c11 + 9c02, is fulfilled. Otherwise, if |3\/3I3| > |C|, then the instability takes place.

£ 0.5-

-0.5-

0.8-

0.6-

0.4-

0.2-

II

0.6

0.8

M

-0.2

-0.4

-0.6

-0.8

-1

[II

LnJ

__________ii

0.97 0.98 0.99 ~7m

II

III

1 e2 L

Fig. 8. Subregions of stability and instability in the case a = a+ = 68"^qq^

Our calculations have shown that in the case a = a- the libration point L1 is stable in Lyapunov sense. In the case a = a+ the there exist three domains of Lyapunov stability in the region of admissible values of parameters i and ^. These domains are separated by two

narrow domains of instability. In Fig. 8 the stability domains are denoted by I, II, and III. The boundaries of the instability domains can be found from the relation |3\/3!3| = \C\. Indeed, substituting (6.10) in the above relation we obtain the following equation for these boundaries

(G+ - 42a/3F+) (G+ + 42A/3F+) = 0. (6.12)

In fact, the boundaries of the instability domains are segments of ellipses intersecting in points El (1412069oV^, and E2 Narrow subregions of instability

denoted by IV, V and highlighted in white are shown on a larger scale at the bottom of Fig. 8.

7. Stability analysis in the case of a first-order resonance

We now investigate the problem of the stability of the libration point L1 for parameter values corresponding to the boundaries of the stability regions, i.e., in cases where one of the following equations is satisfied: a = 1, a = — a = § or a = 0. These cases correspond to first- and second-order resonances. We recall that they occur on two-dimensional surfaces in the three-dimensional space of the parameters i, Q1 and Q2. These surfaces are given by the parametric equations (4.3)-(4.6).

We first consider the case a = 1. In this case, the frequencies of the linear system with the Hamiltonian (3.5) take the following values: u1 = 1 and w2 = 0. One of the frequencies is zero, i.e., a first-order resonance takes place. It is easy to show that at this resonance the matrix of the linear system is not brought to diagonal form. Therefore, the linear system is unstable. However, in this borderline case the linear instability does not imply instability in the complete nonlinear system with the Hamiltonian (3.4). To address the problem of stability in a rigorous manner, a nonlinear analysis is necessary, which we will carry out along the same lines as before.

We first make the linear canonical change of variables

Qi = -x1 + ——y2, q2 = VSx2-2y1,

* (7-1)

p1 = - V3x2 + y1, p2 = x1 - —y2,

which reduces the quadratic part of the Hamiltonian (3.4) to the following normal form:

H2 = \{x\ + yl)--2yl (7.2)

The change of variables (7.1) was obtained by the method described in [19, 20].

We now normalize the part of the Hamiltonian that contains third- and fourth-degree terms in the canonical variables. This can be done using a canonical near-identity change of variables xi,yi ui,vi the generating function of which is sought in the form (5.3). In the case of the first-order resonance considered here, the Hamiltonian can be reduced to the following normal form:

H = ^ (uj + vj)-^v22+k34+k12 (uj + v\) u2+Ai4+B (uj + v\) u22+C (uf + vj)'2+■ ■ ■ . (7.3) The dots in (7.3) and bellow denote terms of degree higher than four.

If the coefficient k3 in the Hamiltonian (7.3) is nonzero, then, according to Sokolskii's theorem [23], the equilibrium point is unstable. To calculate the coefficients of the Hamiltonian (7.3), the Deprit-Hori method was applied. It turned out that in this problem the coefficients k3 and k12 are zero. In this case, conclusions on stability can be drawn by analyzing the fourth-order terms of the Hamiltonian function. It follows from Sokolskii's theorem [23] that, if the inequality A < 0 is satisfied, the equilibrium point x = yi = 0 of the system with the Hamiltonian (7.3) is Lyapunov stable. Otherwise, if A > 0, the equilibrium is unstable. Calculations show that the coefficients of the normalized Hamiltonian (7.3) have the following explicit form:

27 o

A=^(b2-c), B = 9(b2 — c), C = |(62-c). (7.4)

In order to verify the condition for stability, we express the coefficient A in terms of the parameters ^ and To this end we substitute (3.9) and (3.10) into the expression presented above for A and then, in the expression thus obtained, using relations (4.6), replace Q1 and Q2 by their expressions in terms of ^ and This yields

= -1) ,7 ,,

Since ^ <E (0; 1), it follows from (7.5) that A < 0. Hence, the equilibrium point of the system with the Hamiltonian (7.3) is Lyapunov stable. This implies that, for the parameter values on the boundary of the stability region given by Eqs. (4.6), the collinear libration point L1 is Lyapunov

stable.

We now consider the case a = in which the frequencies take the values = and w2 = 0. Here a first-order resonance takes place also. We recall that the boundary (4.3) of the stability region in this case coincides with the bifurcation surface.

We make the linear canonical change of variables

ll):; '2\ 5 \ T5 II)1 \ 5 1 \ 1~5

9l =--25 ' 1 -T-'2' 92 =--—' '/2' (76)

ID1 \ 5 \T5 ll):' \ 5 \T5 1 ' j

=--~2/1 " >'2 = " 5

which reduces the quadratic part of the Hamiltonian (3.4) to the following normal form:

1

H2 = 1T{x2 + y2)--y22. (7.7)

The normalization of the third- and fourth-degree terms, which is performed as in the case a = 1, reduces the Hamiltonian to the form (7.3). The coefficients at the third- and fourth-degree terms of the normalized Hamiltonian have in this case the form

^ = -is-* = (7-8)

108 2 9 „ 9\/IÜ /_„_,9 „„ N _ 88 239,, 801

A =--b2--c, B =----(73162 - 550c) , C =--b2--c. (7.9)

625 25 ' 12 500 v 200 000 4000 v '

Thus, according to Sokolskii's theorem [23], when b = 0, the equilibrium point is unstable.

The coefficient b on the boundary a = — ^ vanishes when

L = + (7.10)

Substituting (7.10) into (4.3), we arrive at the following parametric equations of the curve on which a = — I} and 6 = 0:

J_(2,.+ l)' J_ (2P-3)'

^ 512 1 -fj, ' 512 fj, y '

To draw conclusions on stability for the parameter values on this curve, it is necessary to determine the sign of the coefficient A at the fourth-degree terms of the Hamiltonian function. To this end, substituting (7.11) into the first of Eqs. (7.9), we have

72

A=—----. (7.12)

25(3-2^(2^ + 1) v ;

It follows from (7.12) that A > 0. Hence, by virtue of Sokolskii's theorem [23], the collinear libration point L1 is unstable for a = — ^ and 6 = 0 also. Thus, for all parameter values corresponding to the boundary of the stability region a = — | given by Eqs. (4.3), the collinear libration point L1 is unstable.

8. Stability analysis in the case of a second-order resonance

We now turn to the case a = §, which takes place on the boundary of the stability region given by the parametric equations (4.5). Here the linear system has equal frequencies u1 = w2 = = i. e., a second-order resonance takes place. In this case, using the linear canonical change of variables

Qi = —¿-x2 + —?-Vi> = i-V32/2,

V- r- > t8"1)

Pi = + —V2, P2 = —*2 - —2/1,

the quadratic part of the Hamiltonian (3.4) is brought to the form

H2 = \ + 4) + - x2y{). (8.2)

By the nonlinear canonical near-identity change of variables xi} yi ^ uvi given by a generating function of the form (5.3), the Hamiltonian can be brought to the following normal form [24]:

H = \ iui + + - u2v1)+

+ u + vj) [A u + vl) + B(v2Ui - u2Vi) + C(u1 + v2)] + ... . (8.3)

Calculations performed by the Deprit-Hori method have shown that the coefficients of the normalized Hamiltonian have the form

, 33 291,2 H61 „ 27y/5 , „l2 x ^ 93 069,2 3321 , ^

A =-62--c, B =--— (3876 - 335c) , C =-62--c. 8.4

40 000 1600 ' 10 000 v n 25 000 1000 v '

According to Sokolskii's theorem [25, 26], the equilibrium point of the canonical system with the Hamiltonian (8.3) is Lyapunov stable if the inequality A > 0 is satisfied. Otherwise,

if A < 0, the equilibrium is unstable. We obtain an expression of the coefficient A in terms of and ¡, which, as in the above-mentioned resonant cases, are chosen as parameters of the problem. Sequentially substituting the expressions (3.9), (3.10) and (4.5) into the first of Eqs. (8.4), we have

3 (9272^2 - 1562^ + 263^2 - 9272^ + 781& + 168)

A =

40000(£* + + M - 1)2

From the equation A = 0 we obtain the following equations of the curves that divide the entire region of admissible values of the parameters and i into stability and instability subregions

781 781 5

= 263'U ~ 526 ± 526 ^"292 572At2 + 292 572At + U 329■ (8"5)

These subregions are shown in Fig. 9. In subregions I and II, shown as dark gray, where the inequality A > 0 is satisfied, the collinear libration point L1 is Lyapunov stable. In subregion III shown as light gray, the inequality A < 0 is satisfied, therefore here the collinear libration point L1 is unstable. The boundary curve a, which divides regions I and III, is given by Eq. (8.5), where the sign " +" must be taken on the right-hand side. The boundary curve ff, which divides regions II and III, is given by Eq. (8.5), where the sign " —" must be taken on the right-hand side. To solve the problem of stability on these curves themselves, it is necessary to carry out an additional analysis taking into account terms of at least degree six in the expansion of the Hamiltonian (3.4). Such an analysis is not carried out in this paper.

To complete the stability analysis, we consider the case a = 0, which takes place on the boundary of the stability region given by the parametric equations (4.4). In this case, the second-order resonance u1 = w2 = 1 takes place also, and the Hamiltonian function is reduced to the form (8.3).

The linear change of variables which normalizes the quadratic part of the Hamiltonian has the following form:

91 = — V2, 92 = —V1, P1 = X2, P2 = X1, (8.6)

and the coefficients of the Hamiltonian normalized up to fourth-degree terms are calculated from the formulae

13 L

64 24 ' 12 v ;

Sequentially substituting (3.9), (3.10) and (4.4) into the first of Eqs. (8.7), we have the following expression for the coefficient A in terms of the parameters and

(18ju + 53^ -

64(e,+¡)2(e,+1 — 1)2-

From the equation A = 0 we obtain the following equations for the boundaries of the stability and instability subregions:

el0) = o, =(8.8)

53

In Fig. 10, in the subregions shown as dark gray, the inequality A > 0 is satisfied; therefore, by virtue of the above-mentioned Sokolskii theorem [25, 26], the collinear libration point L1 in these subregions is Lyapunov stable. In the subregion shown as light gray, the inequality A < 0 is satisfied; therefore, here, by virtue of the same theorem, the collinear libration point L1 is unstable.

Fig. 9. Subregions of stability and instability in Fig. 10. Subregions of stability and instability in

the case a = g the case a = 0

Conclusion

In the three-dimensional parameter space of the problem considered there exist two regions in which the collinear libration point L1 is linearly stable. The nonlinear analysis carried out in this work has made it possible to obtain rigorous conclusions on the Lyapunov stability of the collinear libration point L1. In particular, if the system has no third-order resonance, for most parameter values inside the linear stability regions the libration point L1 is Lyapunov stable also. An exception may take place only in the case of degeneracy in which solving the stability problem requires an analysis taking into account terms of degree six or higher in the series expansion of the Hamiltonian in a neighborhood of L1. The values of the parameters that correspond to the case of degeneracy lie on the surface given by the equation A = 0, where A is defined by the formula (5.5).

Inside each of the linear stability regions there exist two-dimensional surfaces corresponding to the third- and fourth-order resonance. On the surfaces corresponding to the third-order resonance the libration point L1 is unstable for most parameter values. An exception may take place only in the special case where the third-degree terms in the expansion of the Hamiltonian of the problem vanish and, despite the presence of a third-order resonance, the libration point L1 is Lyapunov stable. In the three-dimensional parameter space of the problem, to this case

there correspond two curves lying on the two-dimensional surfaces of the third-order resonance. The equations of these curves and the resonant surfaces themselves can be represented in the convenient parametric form (6.1) and (6.5), where the dimensionless coordinate of the libration point L1 and i, which characterizes the ratio between the masses of the attracting bodies are chosen as parameters. On the surfaces corresponding to the fourth-order resonance the libration point L1 is unstable in two narrow subregions of the admissible region of the parameters and L1. Outside these subregions the libration point L1 is stable.

On the boundaries of the linear stability region there exist first- and second-order resonances. Here the libration point L1 can be both stable and unstable in the sense of Lyapunov. Diagrams of stability for the parameter values corresponding to the above-mentioned boundaries are shown in Figs. 8-10.

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