UDC 517.9+521.13

MECHANICS

M. M. KHAPAEV

ON STABILITY IN THE THREE-BODY PROBLEM

(Presented by Academician N. N. Krasovskii on 16 IV 1970)

A generalization of Lyapunov’s second method, proposed in works \((^{1,2})\), makes it possible to carry out a stability investigation in the three-body problem.

We shall write the equations of motion in the three-body problem in the canonical variables \(L, \lambda, \rho, \omega\) \((^{3,4})\)

\[ \frac{d\lambda}{dt}=\frac{\partial F}{\partial L},\qquad \frac{dL}{dt}=-\frac{\partial F}{\partial \lambda}, \]

\[ \frac{d\omega}{dt}=\frac{\partial F}{\partial \rho},\qquad \frac{d\rho}{dt}=-\frac{\partial F}{\partial \omega}, \tag{1} \]

where \(L=m\sqrt{M}\cdot\sqrt{a}\) (\(a\) is the semimajor axis, \(m\) the mass of the planet, \(M\) the mass of the Sun), \(\rho_1=L(1-\sqrt{1-e^2})\) (\(e\) is the eccentricity), \(\rho_2=(L-\rho_1)(1-\cos i)\) (\(i\) is the inclination), \(\lambda\) is the mean longitude, \(\omega_1\) is the longitude of perihelion, \(\omega_2\) is the longitude of the node. The variables \(L_1,\lambda_1,\rho_1,\omega_1,\rho_2,\omega_2\) refer to one planet, \(L_2,\lambda_2,\rho_3,\omega_3,\rho_4,\omega_4\) to the other. \(F=F_0+\mu F_1;\ F_0=-M_1(2L_1^2)^{-1}-M_2(2L_2^2)^{-1}\), where \(M_1\) and \(M_2\) are constants; \(\mu F_1\) is the perturbing function, \(\mu\) is a small parameter having the meaning of the ratio of the planet’s mass to the mass of the Sun; the masses of the planets are assumed to be of the same order of magnitude,

\[ \mu F_1=\sum A\rho_1^{q_1}\rho_2^{q_2}\rho_3^{q_3}\rho_4^{q_4} \cos\left(\sum k_i\lambda_i+\sum p_i\omega_i+h\right). \tag{2} \]

Here \(k_i\) and \(p_i\) are integers over which the summation is performed. \(A\) and \(h\) depend only on \(L\), \(2q_i\) are positive integers, and \(2q_i\ge |p_i|\).

The properties of the perturbing function are studied in detail by H. Poincaré \((^{3,4})\); here we shall only write down the integrals which will be needed below,

\[ \sum L-\sum \rho=\mathrm{const}, \]

\[ \rho_2\left(L_1-\rho_1-\frac{\rho_2}{2}\right) = \rho_4\left(L_2-\rho_3-\frac{\rho_4}{2}\right). \tag{3} \]

We shall denote the coefficients of the series (2) by \(\mu h_{kp}\), and the argument of the \(\cos\) by \(\theta\). We shall also introduce the frequencies \(n_1(L)=M_1L_1^{-3}\) and \(n_2(L)=M_2L_2^{-3}\), assuming that \(n_1\ne n_2\). Equations (1) in these notations are written in the form

\[ \frac{d\lambda}{dt}=n+\mu\sum_{k,p}\frac{\partial h_{kp}}{\partial L}\cos\theta, \]

\[ \frac{dL}{dt}=\mu\sum_{k,p}h_{kp}k\sin\theta, \]

\[ \frac{d\omega}{dt}=\mu\sum_{k,p}\frac{\partial h_{kp}}{\partial \rho}\cos\theta, \]

\[ \frac{d\rho}{dt}=\mu\sum_{k,p}h_{kp}p\sin\theta. \tag{4} \]

The variables \(L,\rho,\omega\) in equations (4) vary slowly—their derivatives are proportional to \(\mu\), and only the two phases \(\lambda\) vary rapidly,

with frequency \(n+O(\mu)\). The vanishing of the combination frequencies \(k_{1}n_{1}(L)+k_{2}n_{2}(L)\) leads to the appearance, in the right-hand sides of equations (4), of slowly varying terms, i.e., to resonance phenomena.

The lines on which the expression \(k_{1}n_{1}(L)+k_{2}n_{2}(L)\) vanishes (resonance lines) are, in the present case, rays issuing from the origin of coordinates in the plane \(L\),

\[ L_{2}/L_{1}=\sqrt[3]{M_{2}/M_{1}}\cdot \sqrt[3]{-k_{2}/k_{1}}. \tag{5} \]

Since the function \(F=F_{0}+\mu F_{1}\) is an integral of the motion, the integral curve in the plane \(L\) remains near the curve \(F_{0}(L)=F_{0}(L_{0})\).

To study stability only with respect to the variables \(L\) of a certain point \(L_{0}\), we use Theorem III of [2], constructing a perturbed Lyapunov function.

Let the point \(L_{0}\) lie on some resonance ray (5) for \(k=k_{0}=\{k_{10},k_{20}\}\). Given \(\varepsilon>0\), we shall indicate such \(\eta(\varepsilon)\), \(T(\varepsilon,\mu)\), and \(\mu_{0}(\varepsilon)\) that a solution in the variables \(L\) satisfying, at the initial moment \(t=0\), the condition \(|L(0)-L_{0}|<\eta\), for all \(0<t<T(\varepsilon,\mu)\) and \(\mu<\mu_{0}\) remains in an \(\varepsilon\)-neighborhood, i.e., \(|L(t)-L_{0}|<\varepsilon\).

The integrals (3) make it possible to conclude that when \(L\) changes within \(O(\varepsilon)\), the variables \(\rho\) undergo changes of the same order and, being small at the initial moment, \(\rho\) remain small for sufficiently small \(\varepsilon\) for \(0<t<T(\varepsilon,\mu)\); therefore there is no need to investigate stability with respect to the variables \(\rho\). It is essential that series of the form (2) in small \(\rho\) will converge rapidly on this time interval of length \(T\). From what has been said above it follows that the integrals (3) determine a certain bound, depending on the initial values of \(\rho\), on the quantity \(\varepsilon\) from above.

Through the point \(L_{0}\) there also passes the curve \(F_{0}(L)=F_{0}(L_{0})\). Introduce a new variable \(x\), measured from the point \(L_{0}\) along the tangent at this point to the curve \(F_{0}(L)=F(L_{0})\); the direction vector of the tangent is \(\mathbf l\). Deviations of the integral curve along the normal to this direction in an \(\varepsilon\)-neighborhood will be small (of order \(o(\varepsilon),\mu\)), since \(F_{0}+\mu F_{1}\) is an integral of the motion; this permits the stability of the point \(L_{0}\) to be investigated only with respect to the single variable \(x\), \(dx=(\mathbf l\cdot d\mathbf L)\).

As the unperturbed Lyapunov function we choose \(v_{0}(L)=|x|\), and we shall seek the perturbed Lyapunov function \(v\) in the form

\[ v=v_{0}(L)+\mu v_{1}(L,\lambda,\rho,\omega,\varepsilon). \tag{6} \]

Differentiating \(v\) by virtue of equations (4), we obtain

\[ \dot v=\mu\sum_{k,p} h_{kp}\cdot (kl)\sin\theta\cdot \operatorname{sgn}x +\mu\,\frac{\partial v_{1}}{\partial\lambda}\left(n+\mu\sum_{k,p}\frac{\partial h_{kp}}{\partial L}\cos\theta\right)+ \]

\[ +\mu^{2}\frac{\partial v_{1}}{\partial L}\sum_{k,p}h_{kp}k\sin\theta +\mu^{2}\frac{\partial v_{1}}{\partial\rho}\sum_{k,p}h_{kp}p\sin\theta +\mu^{2}\frac{\partial v_{1}}{\partial\omega}\sum_{k,p}\frac{\partial h_{kp}}{\partial\rho}\cos\theta. \tag{7} \]

From the series

\[ \dot x\,\operatorname{sgn}x = \mu\sum_{k,p} h_{kp}(kl)\sin\theta\cdot \operatorname{sgn}x \]

we separate all terms whose resonance lines lie in the \(2\varepsilon\)-neighborhood of the point \(L_{0}\), and denote their sum by \(\mu R_{\varepsilon}\). In what follows, the sum of the resonance terms will be denoted by a straight bar over the summation sign, and the sum of the oscillatory terms by a wavy one. The terms of the series belonging to \(k_{0}\) shall be assigned to the oscillatory terms.

Choose some \(\eta>0\) \((\eta<\varepsilon)\) and \(\sigma<\frac12(\varepsilon-\eta)\). We require that the function \(v_{1}\), for \(\eta<|x|<\varepsilon\), satisfy the equation

\[ \frac{\partial v_{1}}{\partial\lambda}\,n = \operatorname{sgn}x\,\widetilde{\sum}_{k,p} h_{kp}(kl)\sin\theta. \tag{8} \]

The denominators \((kn)\) that arise in the integration are bounded below by quantities of order \(\eta\) or \(\varepsilon\). Therefore the function \(v_1\), for \(\eta < |x| < \varepsilon\), is bounded and has order \((O(\varepsilon,\eta))^{-1}\sum |h_{kp}|\). By choosing \(\mu_0\) sufficiently small one can make the perturbation \(\mu v_1\) less than \(\sigma/2\) for all \(\mu < \mu_0\). In this case, from relations (7), taking into account equations (4), it follows that \(\dot v = \mu R_\varepsilon + O(\mu^2)\), and for the time interval \(T\) on which a solution beginning in the \(\eta\)-neighborhood remains in the \(\varepsilon\)-neighborhood, according to Theorem III of paper (²) the estimate is valid

\[ T \sim \frac{\sigma}{2}\,\mu^{-1}[R_\varepsilon+O(\mu)]^{-1}. \]

Let us now estimate the magnitude of the remainder \(R_\varepsilon\), which for fixed \(\varepsilon\) is determined by the rate of convergence of series (2). For this purpose introduce the unit vector \(\chi=\left\{\dfrac{k_i}{|k|}\right\}\) and consider the combination frequency \((\chi n(L))\cdot |k|\). At the point \(L_0\), \(\chi_0 n(L_0)=0\). On other resonance lines \((\chi n)=(\chi_0+\Delta\chi)\cdot(n_0+\Delta n)=0\). On the boundary of the \(2\varepsilon\)-neighborhood, \(\Delta n=O(\varepsilon)\); consequently, for a resonance line lying within the \(2\varepsilon\)-neighborhood, \(\Delta\chi\le O(\varepsilon)\). Thus the smallest \(k\) for which the resonance line will be in the \(2\varepsilon\)-neighborhood is determined by the condition \(k^{-1}=O(\varepsilon)\), and as \(\varepsilon\) decreases the remainder \(R_\varepsilon\), containing all terms of this kind, decreases.

It should be noted that in the concrete three-body problem the parameter \(\mu\) is a small but fixed quantity; therefore the length of the interval \(T\) cannot be increased by decreasing the parameter \(\mu\). However, this can be done by taking advantage of the smallness of \(R_\varepsilon\). It is known (³, ⁴) that the order of a term of series (2) in powers of the small eccentricities and inclinations, belonging to \(k=\{k_1,k_2\}\), is not lower than \(||k_1|-|k_2||\). Suppose that \(R_\varepsilon=O(\mu^2)\), and determine one more approximation for the function \(v\). \(v=v_0(L)+\mu v_1+\mu^2 v_2\), where \(v_1\) has already been determined above. Then

\[ \begin{aligned} \dot v={}&\mu R_\varepsilon+\mu^2\frac{\partial v_1}{\partial\lambda}\frac{\partial F_1}{\partial L} -\mu^2\frac{\partial v_1}{\partial\rho}\frac{\partial F_1}{\partial\omega} +\mu^2\frac{\partial v_1}{\partial\omega}\frac{\partial F_1}{\partial\rho} -\mu^2\frac{\partial v_1}{\partial L}\frac{\partial F_1}{\partial\lambda} \\ &+\mu^2\frac{\partial v_2}{\partial\lambda}\left(n+\mu\frac{\partial F_1}{\partial L}\right) -\mu^3\frac{\partial v_2}{\partial\rho}\frac{\partial F_1}{\partial\omega} +\mu^3\frac{\partial v_2}{\partial\omega}\frac{\partial F_1}{\partial\rho} -\mu^3\frac{\partial v_2}{\partial L}\frac{\partial F_1}{\partial\lambda}. \end{aligned} \tag{9} \]

We carry out here the multiplication of the series containing the derivatives of the function \(v_1\), separate the resonance terms in the manner indicated above, and define \(v_2\) as the solution of the equation

\[ \frac{\partial v_2}{\partial\lambda}n = \frac{\widehat{\partial v_1}}{\partial L}\frac{\partial F_1}{\partial\lambda} -\frac{\widehat{\partial v_1}}{\partial\lambda}\frac{\partial F_1}{\partial L} +\frac{\widehat{\partial v_1}}{\partial\rho}\frac{\partial F_1}{\partial\omega} -\frac{\widehat{\partial v_1}}{\partial\omega}\frac{\partial F_1}{\partial\rho}. \tag{10} \]

Thus the estimate for \(T\) becomes

\[ T(\varepsilon,\mu)\le \frac{\sigma(\varepsilon)} {2\left(\mu R_\varepsilon+O(\mu^3)\right)}. \tag{11} \]

The construction of higher approximations in a concrete problem may prove impossible, since \(\varepsilon\) is bounded below by the number \(\eta\), which must be so large that the \(\eta\)-neighborhood of the resonance line includes the initial conditions of the given problem.

Moscow State University
named after M. V. Lomonosov

Received
20 III 1970

REFERENCES

¹ M. M. Khapaev, DAN, 176, No. 6 (1967).
² M. M. Khapaev, DAN, 193, No. 1 (1970).
³ A. Poincaré, Lectures on Celestial Mechanics, Moscow, 1965.
⁴ H. Poincaré, Les Méthodes nouvelles de la Mécanique céleste, I—III, Paris, 1892, 1893, 1899.