Mathematics

An. M. Leontovich

On the Stability of Lagrangian Periodic Solutions of the Restricted Three-Body Problem

(Presented by Academician A. N. Kolmogorov on 11 XI 1961)

§ 1. Let three material points mutually attract one another according to Newton’s law. In 1772, Lagrange (¹) discovered that the equations of the three-body problem admit equidistant solutions: three bodies placed at the vertices of an equilateral triangle, with appropriate initial velocities, will move in such a way that the triangle formed by them rotates uniformly in its plane about the center of gravity of the bodies. The question naturally arises of the stability of these periodic motions, i.e., of the behavior of the bodies for initial conditions close to equidistant ones. The present work is devoted to the study of this question for the so-called restricted three-body problem (², ³).

In the restricted three-body problem it is assumed that two bodies: \(S\)—of mass \(m_1\), and \(J\)—of mass \(m_2\), move with constant angular velocity \(n\) along circles about their common center of gravity \(O\). The third body (planetoid) \(P\), moving in the plane \(OSJ\), is attracted by them according to Newton’s law, but itself has no influence on the motion of the bodies \(S, J\).

It is known (⁴) that in the first (linear) approximation the equidistant configuration is stable if

\[ (m_1 + m_2)^2 > 27 m_1 m_2 . \tag{1} \]

But it is also known (⁴) that true stability does not necessarily follow from stability in the linear approximation.

In the present work the following is proved.

Theorem 1. For all \(m_1, m_2\) in the domain \((m_1 + m_2)^2 > 27m_1m_2\), except possibly for a set of Lebesgue measure zero, the Lagrangian equidistant solution of the restricted three-body problem is stable.

§ 2. Introduce a rotating system of Cartesian coordinates with origin \(O\) and axis \(Ox = SJ\). Let \(x, y\) be the coordinates of \(P\) in this system; \(u, v\) the projections on the axes \(Ox, Oy\) of the velocity of \(P\) relative to fixed axes. Then the equations of the restricted three-body problem have canonical form with Hamiltonian function (see (⁴))

\[ K = \frac{1}{2}(u^2 + v^2) + n(uy - vx) - \frac{m_1}{SP} - \frac{m_2}{JP}. \tag{2} \]

In the equidistant configuration the planetoid is stationary relative to the rotating system and has constant coordinates \(a, b\). Denote by \(l\) the length \(SJ\). As is known (⁴), \(n^2 l^3 = m_1 + m_2\), and the point \(x = a, y = b, u, v\), where

\[ a = \frac{l}{2}\frac{m_1 - m_2}{m_1 + m_2}, \qquad b = \frac{\sqrt{3}\,l}{2}, \qquad u = -nb, \qquad v = na, \tag{3} \]

is an equilibrium position of the system with Hamiltonian function (2).

Introduce dimensionless time \(\tau = nt\). In a neighborhood of the equilibrium position (3) it is convenient to use the dimensionless variables \(q_1, q_2, p_1, p_2\):

\[ x = a + q_1l,\quad y = b + q_2l,\quad u = -nb + p_1nl,\quad v = na + p_2nl. \]

The derivatives

variables \(q_1, q_2\) and the conjugate variables \(p_1, p_2\) as functions of \(\tau\) are determined from the canonical equations with Hamiltonian function (2), which must be expressed in terms of \(q_1, q_2, p_1, p_2\).

The equilibrium position now has coordinates \(q_1=q_2=p_1=p_2=0\). In a neighborhood of this point the Hamiltonian function can be expanded in a series in powers of \(q_1, q_2, p_1, p_2\). Omitting one constant term, we obtain

\[ \begin{aligned} K=K_2+K_3+\cdots+K_m+\cdots &= \frac12 (C\mathbf r,\mathbf r) + \sum_{\nu_1+\nu_2+\nu_3+\nu_4=3} f_{\nu_1\nu_2\nu_3\nu_4} q_1^{\nu_1}q_2^{\nu_2}p_1^{\nu_3}p_2^{\nu_4} + \cdots , \end{aligned} \tag{4} \]

where \(\mathbf r=(q_1,q_2,p_1,p_2)\); \(C=\|c_{ij}\|\); \(i,j=1,2,3,4\); \(K_m\) denotes the collection of all terms of degree \(m\) with respect to \(q_1,q_2,p_1,p_2\).

In particular, for the Hamiltonian function (2), introducing the parameter

\[ k=\frac{3\sqrt3}{4}\frac{m_1-m_2}{m_1+m_2}, \]

it is not difficult to find that

\[ K_2=\frac12 p_1^2+\frac12 p_2^2+q_2p_1-q_1p_2+\frac18 q_1^2-kq_1q_2-\frac58 q_2^2, \tag{5} \]

\[ K_3=-\frac{7\sqrt3 k}{36}q_1^3+\frac{3\sqrt3}{16}q_1^2q_2+\frac{11\sqrt3 k}{12}q_1q_2^2+\frac{3\sqrt3}{16}q_2^3, \tag{5'} \]

\[ K_4=\frac{37}{128}q_1^4+\frac{25k}{24}q_1^3q_2-\frac{123}{64}q_1^2q_2^2-\frac{15k}{8}q_1q_2^3-\frac{3}{128}q_2^4. \tag{5''} \]

Let us introduce, instead of \(k\), the parameter \(\Delta=(27/16-k^2)^{-1}\). Then the Hamiltonian function (2) depends on the single parameter \(\Delta\).

§ 3. The canonical system of equations with Hamiltonian function (2) is linear:

\[ \dot{\mathbf r}=C\mathbf r,\qquad \mathbf r=(q_1,q_2,p_1,p_2);\qquad C=\|c_{ij}\|;\quad i,j=1,2,3,4. \]

The characteristic equation of this system is, according to (5),

\[ \lambda^4+\lambda^2+\Delta^{-1}=0 \tag{6} \]

and under condition (1), i.e. if \(\Delta>4\), it has purely imaginary roots \(\pm\lambda_1,\pm\lambda_2\).

It is known \({}^{(2)}\) that the quadratic form \(K_2\) can be reduced by a complex canonical linear transformation to normal form, i.e. there exists a symplectic matrix \(A\), \(A=\|a_{ij}\|\); \(i,j=1,2,3,4\), such that if we make the substitution \(\mathbf r=A\bar{\mathbf r}\), where \(\mathbf r=(q_1,q_2,p_1,p_2)\), \(\bar{\mathbf r}=(\bar q_1,\bar q_2,\bar p_1,\bar p_2)\), then the Hamiltonian function (4) takes the form:

\[ K=\lambda_1\bar q_1\bar p_1+\lambda_2\bar q_2\bar p_2 \tag{7} \]

\[ \quad+ \sum_{\nu_1+\nu_2+\nu_3+\nu_4=3} g_{\nu_1\nu_2\nu_3\nu_4} \bar q_1^{\nu_1}\bar q_2^{\nu_2}\bar p_1^{\nu_3}\bar p_2^{\nu_4} + \sum_{\nu_1+\nu_2+\nu_3+\nu_4=4} h_{\nu_1\nu_2\nu_3\nu_4} \bar q_1^{\nu_1}\bar q_2^{\nu_2}\bar p_1^{\nu_3}\bar p_2^{\nu_4} +\cdots . \]

The corresponding transformation \((q_1,q_2,p_1,p_2)\to(\bar q_1,\bar q_2,\bar p_1,\bar p_2)\) will be canonical.

Furthermore, it is known \({}^{(5)}\) that if

\[ \lambda_2/\lambda_1\ne 0,\ \pm \tfrac13,\ \pm \tfrac12,\ \pm 1,\ \pm 2,\ \pm 3,\ \infty, \]

then by means of a complex analytic canonical transformation
\[ (\bar q_1,\bar q_2,\bar p_1,\bar p_2)\to(\widetilde q_1,\widetilde q_2,\widetilde p_1,\widetilde p_2) \]
one can bring the Hamiltonian function (7) to the form:

\[ K=\lambda_1\widetilde q_1\widetilde p_1+\lambda_2\widetilde q_2\widetilde p_2 +\alpha \widetilde q_1^{\,2}\widetilde p_1^{\,2} +\beta \widetilde q_1\widetilde q_2\widetilde p_1\widetilde p_2 +\gamma \widetilde q_2^{\,2}\widetilde p_2^{\,2} +\cdots , \tag{8} \]

where the terms of order higher than the fourth with respect to \(\widetilde q_1,\widetilde q_2,\widetilde p_1,\widetilde p_2\) have not been written. Here \(\lambda_1,\lambda_2,\alpha,\beta,\gamma\) are analytic functions of \(\Delta\) for \(\Delta>0\).

Using the standard method of reducing the Hamiltonian function (7) to the normal form (8) (see (⁵)), one can obtain the following expressions for \(\alpha,\beta,\gamma\):

\[ \begin{aligned} \alpha={}&-\frac{3}{\lambda_1}g_{3000}g_{0030} -\frac{3}{\lambda_1}g_{2010}g_{1020} +\frac{1}{2\lambda_1-\lambda_2}g_{0120}g_{2001} -\frac{1}{\lambda_2}g_{1110}g_{1011} \\ &-\frac{1}{2\lambda_1+\lambda_2}g_{2100}g_{0021} +h_{2020}; \\[4pt] \beta={}&\frac{4}{\lambda_1-2\lambda_2}g_{0210}g_{1002} -\frac{4}{\lambda_1+2\lambda_2}g_{1200}g_{0012} -\frac{4}{2\lambda_1+\lambda_2}g_{2100}g_{0021} \\ &-\frac{4}{2\lambda_1-\lambda_2}g_{2001}g_{0120} -\frac{2}{\lambda_1}g_{2010}g_{0111} -\frac{2}{\lambda_1}g_{1101}g_{1020} \\ &-\frac{2}{\lambda_2}g_{0201}g_{1011} -\frac{2}{\lambda_2}g_{1110}g_{0102} +h_{1111}; \\[4pt] \gamma={}&-\frac{3}{\lambda_2}g_{0300}g_{0003} -\frac{3}{\lambda_2}g_{0102}g_{0201} -\frac{1}{\lambda_1-2\lambda_2}g_{1002}g_{0210} \\ &-\frac{1}{\lambda_1}g_{1101}g_{0111} -\frac{1}{\lambda_1+2\lambda_2}g_{1200}g_{0012} +h_{0202}. \end{aligned} \tag{9} \]

In (⁶) a sufficient condition for stability of the equilibrium position was found. In our notation it can be formulated as follows:

Let the Hamiltonian function (4) be such that:

I. \(\lambda_1\) and \(\lambda_2\) are purely imaginary.
II. \(\lambda_1/\lambda_2 \notin \mathfrak{M}\), where \(\mathfrak{M}\) is a certain set of measure zero on the line.
III. The inequality \(\Phi \equiv \alpha\lambda_2^2-\beta\lambda_1\lambda_2+\gamma\lambda_1^2 \ne 0\) is satisfied.

Then the equilibrium position \(q_1=q_2=p_1=p_2=0\) is indeed stable.

Our theorem will be proved if we verify that conditions II and III are satisfied for almost all \(\Delta>4\).

Note that \(\Phi\) and \(\lambda_1/\lambda_2\) are analytic functions of one variable \(\Delta\) for \(\Delta>0\). From (6) it is seen that \(\lambda_1/\lambda_2\) is not constant; therefore the set of those \(\Delta\) for which \(\lambda_1/\lambda_2\) belongs to \(\mathfrak{M}\) has measure zero. Further, if \(\Phi\) does not vanish identically, then condition III is satisfied for almost all \(\Delta\). Thus, to prove the theorem it is enough to verify that \(\Phi\not\equiv0\).

§ 4. For this purpose we shall prove that \(|\Phi|\to\infty\) as \(\Delta\to\infty\) (i.e., when \(m_2/m_1\to0\), \(k^2\to{}^{27}/_{16}\)). We shall prove this starting from a more general lemma, to the formulation of which we now proceed.

Introduce some notation. Let \(B=\|b_{ij}\|=\bar C^2\). It is easy to verify that

\[ B= \begin{pmatrix} b_{11} & b_{12} & 0 & b_{14}\\ b_{21} & b_{22} & -b_{14} & 0\\ 0 & -b_{41} & b_{11} & b_{21}\\ b_{41} & 0 & b_{12} & b_{22} \end{pmatrix}. \]

Next, introduce the matrix

\[ \widetilde B= \begin{pmatrix} N_2 & -b_{12} & -b_{12} & N_1\\ b_{21} & N_2 & N_1 & b_{21}\\ 0 & b_{41} & b_{41} & 0\\ b_{41} & 0 & 0 & b_{41} \end{pmatrix} =\|\widetilde b_{ij}\|,\qquad i,j=1,2,3,4, \]

where \(N_1=b_{11}-\lambda_1^2=-b_{22}+\lambda_2^2\), \(N_2=b_{11}-\lambda_2^2=-b_{22}+\lambda_1^2\), and put
\[ D=\|d_{ij}\|=\widetilde B' C \widetilde B \]
(see (4)), where \(\widetilde B'\) is the matrix transposed to \(\widetilde B\).

Let, in the Hamiltonian function (4), the coefficients \(c_{ij}\) and \(f_{\nu_1\nu_2\nu_3\nu_4}\) be analytic functions of \(\Delta\) for \(\Delta>0\). Suppose that \(c_{ij}\) and \(f_{\nu_1\nu_2\nu_3\nu_4}\), as \(\Delta\to\infty\), have finite limits \(C_{ij}\) and \(F_{\nu_1\nu_2\nu_3\nu_4}\). Then \(c_{ij}, b_{ij}, \widetilde b_{ij}, d_{ij}, \lambda_1,\lambda_2\) are analytic functions of \(\Delta\) for \(\Delta>0\), having as \(\Delta\to\infty\) the limits \(C_{ij}, B_{ij}, \widetilde B_{ij}, D_{ij}, \Lambda_1,\Lambda_2\), respectively.

We shall consider the case when

\[ \Lambda_1=0,\qquad \text{but } \lambda_1\ne 0;\qquad \Lambda_2\ne 0. \tag{10} \]

Then the following is true.

Lemma. Suppose condition (10) is satisfied, as well as the conditions

A. \(B_{41}\ne 0\).

B. \(D_{11}\ne 0\).

C.

\[ \sum_{\nu_1+\nu_2+\nu_3=3} F_{\nu_1\nu_2\nu_3 0}\,\widetilde B_{i_1 3}\widetilde B_{i_2 3}\widetilde B_{i_3 3}\ne 0, \]

where \(1\le i_1\le i_2\le i_3\le 3\), and in the set \((i_1,i_2,i_3)\) there are \(\nu_1\) ones, \(\nu_2\) twos, and \(\nu_3\) threes.

Then \(\Phi\ne 0\).

§ 5. Let us briefly outline the proof of the lemma.

\(1^\circ\). Computing the matrix \({}^*A\), by means of the usual, rather lengthy, calculations we obtain **

\[ a_{ij}=O(1),\qquad \text{if } j=2,4 \text{ or if } i=4; \tag{11} \]

\[ a_{i1}\sim M|\lambda_1|^{-1/2}\widetilde b_{i3},\qquad a_{i3}\sim -M|\lambda_1|^{-1/2}\widetilde b_{i3}\quad (i=1,2,3), \tag{12} \]

where \(M\) is a certain complex constant independent of \(i\).

\(2^\circ\). Using (9) and condition III, it is easy to obtain an expression for \(\Phi\). For us it will be important that \(\Phi=\Phi_1+\Phi_2+\Phi_3\), where

\[ \Phi_1=\lambda_1^2 h_{0202}-\lambda_1\lambda_2 h_{1111}+\lambda_2^2 h_{2020}, \]

\[ \Phi_2=-\frac{3\lambda_2^2}{\lambda_1}g_{3000}g_{0030} -\frac{3\lambda_2^2}{\lambda_1}g_{2010}g_{1020} =-\frac{3\lambda_2^2}{\lambda_1}\widetilde\Phi_2, \]

and \(\Phi_3\) is the sum of 16 terms of the form

\[ \frac{r_3\lambda_1^\delta\lambda_2^{\,2-\delta}} {r_1\lambda_1+r_2\lambda_2}\, g_{\nu_1\nu_2\nu_3\nu_4}\, g_{\nu'_1\nu'_2\nu'_3\nu'_4} \]

\[ (\delta=0,1,2;\ r_1=0,\pm1,\pm2;\ r_2=\pm1,\pm2;\ r_3=1,2,3,4). \]

\(3^\circ\). Let \(\Delta\to\infty\). Using (11), (12), and the fact that in the preceding formula \(r_2\ne 0\), it is easy to obtain that each of the 19 terms in \(\Phi_1+\Phi_3\) is \(O(|\lambda_1|^{-3})\). Further, since condition C is satisfied, \(g_{3000}\succ |\lambda_1|^{-3/2}\). Then, using (12), it is quite simple to show that \(g_{3000}\sim -g_{0030}\). By analogous considerations, either \(g_{2010}=o(|\lambda_1|^{-3/2})\), or \(g_{2010}\succ |\lambda_1|^{-3/2}\) and \(g_{2010}\sim -g_{1020}\). Therefore from (12) it follows easily that \(\widetilde\Phi_2\succ |\lambda_1|^{-3}\). Hence \(\Phi_2\succ |\lambda_1|^{-4}\). Since \(\Phi_1+\Phi_3=O(|\lambda_1|^{-4})\), it follows that \(\Phi\succ |\lambda_1|^{-4}\), i.e. \(|\Phi|\to\infty\) as \(\Delta\to\infty\). Thus, \(\Phi\ne 0\), as was required to prove.

§ 6. It remains to prove the theorem. Using expressions (5), (5′), and (6), it is quite simple to show that the Hamiltonian function (2) satisfies the conditions of the lemma. Consequently, \(\Phi\ne 0\). Hence, according to the remark at the end of § 3, the theorem follows.

In conclusion I express my deep gratitude to V. I. Arnold for formulating the problem and for his constant help in carrying it out and in preparing the article for print.

Received
10 XI 1961

REFERENCES

1. J. L. Lagrange, Oeuvres, 6, Paris, 1873, pp. 272–292.
2. К. Л. Зигель, Лекции по небесной механике, Moscow, 1959.
3. J. E. Littlewood, Proc. London Math. Soc., 3, 9, No. 35, 343 (1959).
4. E. T. Уиттекер, Аналитическая динамика, Moscow–Leningrad, 1937, Ch. XV.
5. Дж. Д. Биркгоф, Динамические системы, Moscow–Leningrad, 1941, Ch. III.
6. В. И. Арнольд, DAN, 137, No. 2, 255 (1961).

\[ \text{* It should be noted that the matrix } A \text{ is not determined uniquely; but there exists one for which relations (11) and (12) hold.} \]

\[ \begin{aligned} \text{** } \varphi_1(\Delta)&=O(\varphi_2(\Delta)), &&\text{if } \varlimsup_{\Delta\to\infty}\left|\varphi_1(\Delta)/\varphi_2(\Delta)\right|<\infty;\\ \varphi_1(\Delta)&=o(\varphi_2(\Delta)), &&\text{if } \lim_{\Delta\to\infty}\varphi_1(\Delta)/\varphi_2(\Delta)=0;\\ \varphi_1(\Delta)&\sim \varphi_2(\Delta), &&\text{if } \lim_{\Delta\to\infty}\varphi_1(\Delta)/\varphi_2(\Delta)=1;\\ \varphi_1(\Delta)&\succ \varphi_2(\Delta), &&\text{if } 0<\varliminf_{\Delta\to\infty}\left|\varphi_1(\Delta)/\varphi_2(\Delta)\right| \le \varlimsup_{\Delta\to\infty}\left|\varphi_1(\Delta)/\varphi_2(\Delta)\right|<\infty. \end{aligned} \]