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MATHEMATICS
V. I. ARNOLD
ON THE STABILITY OF AN EQUILIBRIUM POSITION OF A HAMILTONIAN SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS IN THE GENERAL ELLIPTIC CASE
(Presented by Academician A. N. Kolmogorov, 9 XII 1960)
§ 1. Let the point \(p=q=0\) be a stationary point of the system
\[ \dot q=\frac{\partial H}{\partial p},\qquad \dot p=-\frac{\partial H}{\partial q}, \tag{1} \]
where \(H(p,q,t)\) is a function analytic in \(p,q,t\), periodic in \(t\) with period \(2\pi\). The case is called elliptic when this equilibrium position is stable in the first (linear) approximation. Then, as Birkhoff showed \((^{1})\), for a suitable choice of the variables \(p,q\), the Hamiltonian function has the form
\[ H=\lambda r+c_2r^2+\cdots+c_nr^n+\widetilde H(p,q,t), \tag{2} \]
where \(2r=p^2+q^2\), \(\widetilde H=O(r^{n+1})\) is an analytic function of \(p,q,t\), and \(n\ge 2\) is arbitrary. We shall call the case general elliptic when among the constants \(c_l\) \((2\le l<\infty)\) there is one different from zero.
§ 2. Examples are known in which the equilibrium position is unstable and \(\lambda\) is rational \((^{2})\). We shall consider the case of irrational \(\lambda\). Denote by \(\Lambda_K\) the set of such \(\lambda\) for which the inequalities
\[ |\lambda n-m|>\frac{K}{(|m|+|n|)^2} \tag{3} \]
are satisfied for all integers \(m,n>0\). Denote by \(\Lambda\) the union of the points of density of all sets \(\Lambda_K\). As is known, the complement of \(\Lambda\) on the line has measure zero \((^{3})\).
Theorem 1. If \(\lambda\in\Lambda\), then the equilibrium position \(0,0\) of the system of equations (1) with Hamiltonian function \(H(p,q,t)\) of general elliptic type (2) is stable.
Theorem 2. Under the hypotheses of Theorem 1, in any neighborhood of the circle \(p=q=0\) of the space \(p,q,t\) there exists an analytic invariant torus \(T_\mu\) with equation \(r=r(\varphi,t)\) \(\left(\varphi=\arctg\frac{p}{q}\right)\). On the torus \(T_\mu\) one can introduce an analytic coordinate \(\psi(\varphi,t)\) such that equations (1) on the torus \(T_\mu\) take the form \(\dot\psi=\mu\). The set formed by the tori \(T_\mu\) has positive measure in the space \(p,q,t\).
Theorem 3. Let the Hamiltonian function have the form
\[ H(r,\varphi,t)=H_0(r)+\widetilde H(r,\varphi,t), \tag{4} \]
where \(dH_0/dr=\mu+\Omega(r)\), \(\mu\in\Lambda_K\), \(\Omega(0)=0\), and the function
\[ \widetilde H=\sum_{m^2+n^2\ne0} H_{mn}(r)e^{i(m\varphi+nt)} \]
for \(|\operatorname{Im}\varphi,t|\le \rho,\ |r|\le \rho_r=\delta^k\) is analytic and satisfies the inequality
\[ |\widetilde H|\le M=\delta^N, \tag{5} \]
and the function \(\Omega(r)\), for \(|r|\leqslant \rho_r\), is analytic and
\[ \delta^a=\theta\leqslant \left|\frac{d\Omega}{dr}\right|\leqslant \Theta=\delta^{-b}. \]
Here \(\delta>0\) is a certain constant; \(N,k,a,b\) are natural numbers. If the inequalities
\[ \begin{gathered} 2k+28+2a+4b<N<3k-14-2b;\\ \delta<10^{-6}K^2;\quad \delta<0.1\,\rho, \end{gathered} \tag{7} \]
are satisfied, then there exist functions \(R(\varphi,t)\), \(\Psi(\varphi,t)\), of period \(2\pi\) in \(\varphi\) and \(t\), analytic for \(|\operatorname{Im}\varphi,t|\leqslant 0.1\,\rho\), and such that on the torus \(r=R(\varphi,t)\), from the equations
\[ \dot\varphi=\frac{\partial H}{\partial r},\qquad \dot r=-\frac{\partial H}{\partial \varphi} \]
it follows that \(\dot\psi=\mu\) (here \(\psi=\varphi+\Psi\)).
Theorem 1 follows from Theorem 2, since the tori \(T_\mu\) separate the circle \(r=0\) from the remaining part of the space \(p,q,t\). Theorem 2 follows from Theorem 3: it is not difficult to see that, under the hypotheses of Theorem 2, there exist arbitrarily small toroidal rings \(|r-r_0|\leqslant \rho_r\) around the circle \(r=0\) to which Theorem 3 is applicable, if in it one takes \(r-r_0\) as the variable \(r\).
§ 3. The last two theorems are generalized to systems with \(n\) degrees of freedom. However, the resulting invariant \((n+1)\)-dimensional tori do not divide the \((2n+1)\)-dimensional phase space \(p,q,t\), and the question of stability remains open. Analogous theorems can also be proved concerning a neighborhood of the equilibrium position of an autonomous Hamiltonian system. In this case, in the \((2n-1)\)-dimensional manifolds \(H(p,q)=h\) there lie tori of dimension \(n\). Hence it follows:
Theorem 4. The equilibrium position of an autonomous Hamiltonian system of equations with two degrees of freedom in the general elliptic case is stable, if \(\lambda_2/\lambda_1\in\Lambda\).
By the general elliptic case we mean here the case when the analytic function \(H\), in suitable coordinates, has the form \((^1)\)
\[ H(p_1,p_2,q_1,q_2)=\lambda_1 r_1+\lambda_2 r_2+H_0(r_1,r_2)+\widetilde H(p_1,p_2,q_1,q_2), \]
where
\[ H_0(r_1,r_2)=\sum_{i+j=2}^{n} c_{ij}r_1^i r_2^j,\qquad \widetilde H=O(r_1+r_2)^{n+1},\qquad 2r_i=p_i^2+q_i^2 \]
and \(h(\varepsilon)=H_0(\varepsilon\lambda_2,-\varepsilon\lambda_1)\) does not vanish identically.
One can also show that any analytic canonical mapping of the plane onto itself near a fixed point of general elliptic type is stable, if its rotation number \(\lambda\in\Lambda\). Theorems 2 and 3 admit a corresponding generalization even in the multidimensional case.
§ 4. We shall now outline the proof of Theorem 3. It is a strengthening of A. N. Kolmogorov’s theorem on the preservation of conditionally periodic motions under a small change of the Hamiltonian function \((^4)\). The invariant torus is found, as in A. N. Kolmogorov’s work, by successive approximations of Newton-method type. This method gives such rapid convergence that it cannot be destroyed by the small divisors appearing in formula (9).
Basic Lemma. Under assumptions (4)—(7) of Theorem 3, there exist an analytic function
\[ \widetilde F(\bar r,\varphi,t)=\sum_{m^2+n^2\ne 0} F_{mn}(\bar r)e^{i(m\varphi+nt)} \]
and a number \(\bar r^*\) such that the canonical transformation
\[ \bar\varphi=\varphi+\partial\widetilde F/\partial\bar r,\qquad \bar r=\bar r'-\bar r^*,\qquad r=\bar r'+\partial\widetilde F/\partial\varphi \]
brings the Hamiltonian function (4) to the form \(\bar H(\bar r,\bar\varphi,t)=\bar H_0+\tilde{\bar H}\), where \(d\bar H_0/d\bar r=\mu+\bar\Omega(\bar r)\), \(\bar\Omega(0)=0\), and the function
\[ \tilde{\bar H}(\bar r,\bar\varphi,t)= \sum_{m^2+n^2\ne 0}\bar H_{mn}(\bar r)e^{i(m\bar\varphi+nt)} \]
is analytic for \(|\operatorname{Im}\bar\varphi,t|\le \bar\rho=\rho-3\delta,\ |\bar r|\le \bar\rho_r=\delta^k\), and satisfies the inequality
\[ |\tilde{\bar H}|\le \bar M=\delta^N, \]
while \(\bar\Omega(\bar r)\) is analytic for \(|\bar r|\le \bar\rho_r\) and
\[ \delta^a=\bar\theta\le \left|\frac{d\bar\Omega}{d\bar r}\right|\le \bar\Theta=\delta^{-b}. \]
In these formulas \(\bar\delta=\delta^{1^{1}/_{2}}\).
Theorem 3 is derived from the fundamental lemma without special difficulty, since the error of the \(s\)-th approximation \(M_s\) is not greater than \(M^{(1^{1}/_{2})^s}\).
Not being able to give here the proof of the fundamental lemma, I shall indicate only the method of constructing \(\tilde F\) and \(\bar r^*\). As is known, \(\bar H(\bar r,\bar\varphi,t)=H'(\bar r',\varphi,t)=\hat H(\bar r',\varphi,t)\), where
\[ \hat H(\bar r',\varphi,t)=H(r(\bar r',\varphi,t),\varphi,t)+ \frac{\partial\tilde F(\bar r',\varphi,t)}{\partial t}. \]
Obviously,
\[ \hat H(\bar r',\varphi,t)=H_0(\bar r')+\hat S_1+\hat S_2+\hat S_3, \]
where
\[ \hat S_1(\bar r',\varphi,t)= \mu\frac{\partial\tilde F}{\partial\varphi} +\frac{\partial\tilde F}{\partial t} +\tilde H; \]
\[ \hat S_2(\bar r',\varphi,t)= H_0(\bar r)-H_0(\bar r')-\mu(r-\bar r'),\qquad |\hat S_2|=|\Omega|\left|\frac{\partial\tilde F}{\partial\varphi}\right|; \tag{8} \]
\[ \hat S_3(\bar r',\varphi,t)= \tilde H(\bar r)-\tilde H(\bar r'),\qquad |\hat S_3|=\left|\frac{\partial\tilde H}{\partial r}\right| \left|\frac{\partial\tilde F}{\partial\varphi}\right|. \]
The function \(\tilde F\) is determined from the condition \(\hat S_1\equiv 0\):
\[ F_{mn}=\frac{iH_{mn}}{\mu m+n}. \tag{9} \]
Passing to the variables \(\bar r,\bar\varphi,t\), we find
\[ \bar H'(\bar r,\bar\varphi,t)= \bar H'_0(\bar r)+\tilde S_{2_0}(\bar r)+\tilde S_{3_0}(\bar r)+ \tilde{\bar H}'(\bar r,\bar\varphi,t) =\tilde{\bar H}_0+\tilde{\bar H}', \]
where \(\tilde{\bar H}'(\bar r,\bar\varphi,t)=\tilde S_2+\tilde S_3\) combines the variable terms of the Fourier series in \(\bar\varphi,t\) of the functions
\[ S_i(\bar r,\bar\varphi,t)=\tilde S_i+S_{i_0}(\bar r) =\hat S_i(\bar r,\varphi(\bar r,\bar\varphi,t),t),\qquad (i=2,3). \]
Now \(\bar r^*\) is determined from the equation
\[ \left.\frac{d\tilde{\bar H}_0}{d\bar r'}\right|_{\bar r^*}=\mu. \]
In this, to estimate \(\bar r^*\) one uses inequality (6).
When inequalities (7) are satisfied, the quantity \(\tilde{\bar H}\), estimated by formulas (8), does not exceed \(M^{1^{1}/_{2}}=\bar M\).
Moscow State University
named after M. V. Lomonosov
Received
26 XI 1960
CITED LITERATURE
- G. D. Birkhoff, Dynamical Systems, Moscow, 1941, Ch. III.
- T. Levi-Civita, Ann. Math. Pura. Appl., (3), 5, 221 (1901).
- A. Ya. Khinchin, Continued Fractions, Moscow, 1949, § 14.
- A. N. Kolmogorov, DAN, 98, No. 4, 527 (1954).