V. K. MELNIKOV

ON SOME CASES OF PRESERVATION OF CONDITIONALLY PERIODIC MOTIONS UNDER A SMALL CHANGE OF THE HAMILTONIAN FUNCTION

(Presented by Academician A. N. Kolmogorov, May 3, 1965)

In the present note there is contained the following theorem, generalizing A. N. Kolmogorov’s theorem \((^{1,2})\) on the preservation of conditionally periodic motions under a small change of the Hamiltonian function.

Theorem. Consider the Hamiltonian system

\[ \dot{x}=-\partial H/\partial y,\qquad \dot{y}=\partial H/\partial x \qquad (x=(x_1,\ldots,x_m),\ y=(y_1,\ldots,y_m)), \]

\[ \dot{p}=-\partial H/\partial q,\qquad \dot{q}=\partial H/\partial p \qquad (p=(p_1,\ldots,p_n),\ q=(q_1,\ldots,q_n)) \tag{1} \]

with Hamiltonian function of the form

\[ H=H_0(x,y,p)+H_1(x,y,p,q), \]

possessing the following properties:

1. The function \(H_0(x,y,p)\) depends analytically on \(x,y,p\) in the domain
   \[ |x|<h,\quad |y|<h,\quad p\in G \quad (h>0). \]

2. \(\partial H_0/\partial x=\partial H_0/\partial y=0\) for \(x=y=0\).

3. The functional determinant of the matrix

\[ \mathcal H_0= \begin{vmatrix} \partial^2 H_0/\partial x\,\partial y & -\partial^2 H_0/\partial x^2\\ \partial^2 H_0/\partial y^2 & -\partial^2 H_0/\partial y\,\partial x \end{vmatrix} \]

does not vanish in the domain \(G\) for \(x=y=0\)*.

1. The functional determinant \(\det|\partial^2 H_0/\partial p^2|\) does not vanish in the domain \(G\) for \(x=y=0\).

2. In the domain \(G\) the conditions

\[ \lambda_\alpha(p)-i(\omega,k)\ne 0,\qquad \lambda_\alpha(p)+\lambda_\beta(p)-i(\omega,k)\ne 0, \]

are satisfied, where \(\lambda_\alpha(p)\) and \(\lambda_\beta(p)\) are arbitrary eigenvalues of the matrix \(\mathcal H_0\) for \(x=y=0\), \(\omega=\omega(p)=\partial H_0/\partial p|_{x=y=0}\), \((\omega,k)=\sum_{j=1}^{n}\omega_j k_j\), and \(k_j\) are arbitrary integers.

1. The function \(H_1(x,y,p,q)\) is analytic in \(x,y,p,q\) in the domain \(F\):
   \[ |x|<h,\quad |y|<h,\quad p\in G,\quad |\operatorname{Im} q|<\rho \]
   and is periodic in \(q\) with period \(2\pi\) \((\rho>0)\).

Then for any \(\chi>0\) there exists an \(M=M(\chi,\rho,h,G,H_0)>0\) such that if in the domain \(F\): \(|x|<h,\ |y|<h,\ p\in G,\ |\operatorname{Im}q|<\rho\), the inequality
\[ |H_1(x,y,p,q)|<M \]
holds, then there exists a decomposition \(\tilde G=G_1\cup G_2\) of the domain \(\tilde G=\operatorname{Re}G\)** such that \(G_2\) is small, i.e. \(\operatorname{mes}G_2\le \chi\,\operatorname{mes}\tilde G\), and for every \(p\in G_1\) there exists a torus \(T_{p\omega}\), invariant with respect to the motions of system (1), possessing the following properties:

* This condition is superfluous if \(\partial H_1/\partial x=\partial H_1/\partial y=0\) for \(x=y=0\).

** \(\tilde G=\operatorname{Re}G\) is the intersection of the domain \(G\) with the subspace \(\operatorname{Im}p=0\); here the domain \(G\) is assumed to be bounded.

1. The invariant tori \(T_{p\omega}\) are given by the parametric equations

\[ x=f_{\omega}(Q),\qquad y=g_{\omega}(Q),\qquad p=p_{\omega}+h_{\omega}(Q),\qquad q=Q+r_{\omega}(Q), \]

where \(f_{\omega}(Q), g_{\omega}(Q), h_{\omega}(Q)\), and \(r_{\omega}(Q)\) are analytic functions of \(Q\), periodic in \(Q\) with period \(2\pi\).

1. The invariant tori \(T_p\) differ little from the unperturbed tori \(x=y=0,\ p=p_{\omega}\), i.e.,

\[ |f_{\omega}(Q)|<\chi,\qquad |g_{\omega}(Q)|<\chi,\qquad |h_{\omega}(Q)|<\chi,\qquad |r_{\omega}(Q)|<\chi . \]

1. The motion on the torus \(T_{p\omega}\) is conditionally periodic with \(n\) frequencies \(\omega=(\omega_1,\ldots,\omega_n)\), i.e. \(\dot Q=\omega\), where \(\omega=\partial H_0/\partial p\big|_{x=y=0,\ p=p_\omega}\).

The proof of this theorem is based on the possibility of transforming system (1), by means of a canonical change of variables, into a Hamiltonian system with Hamiltonian function of the form

\[ H=\bar H(X,Y,P)+H_2(X,Y,P,Q); \]

where \(\partial \bar H/\partial X=\partial \bar H/\partial Y=0\) for \(X=Y=0\), and the function \(H_2(X,Y,P,Q)\) in the domain \(F'\): \(|X|<h'<h,\ |Y|<h',\ P\in G'\subset G,\ |\operatorname{Im} Q|<\rho'<\rho\) (\(h'>0,\rho'>0\)) satisfies the inequality

\[ |H_2(X,Y,P,Q)|<M^{1+\frac16}, \tag{2} \]

and the measure of the difference \(G\setminus G'\) is small together with \(M\).

For this purpose put

\[ H(x,y,p,q)=\bar H(x,y,p)+\tilde H(x,y,p,q), \tag{3} \]

where

\[ \bar H(x,y,p)=H_0(x,y,p)+\bar H_1(x,y,p), \]

\[ \bar H_1(x,y,p)=(2\pi)^{-n}\oint H_1(x,y,p,q)\,dq \]

and the contour integral in the last equality is taken over the surface of the \(n\)-dimensional torus. Let, further, \(x=f(p);\ y=g(p)\) be the solution of the system of equations \(\partial \bar H/\partial x=\partial \bar H/\partial y=0\), which reduces to the solution \(x=y=0\) when \(\bar H_1(x,y,p)\equiv 0\). We now perform in system (1) the canonical change of variables (3)

\[ x=x'+f(p'),\qquad y'=y-g(p'),\qquad p=p',\qquad q'=q+\frac{\partial f}{\partial p'}y-\frac{\partial g}{\partial p'}x' \]

with generating function \(V=x'y+p'q+f(p')y-g(p')x'\). As a result of the change of variables the Hamiltonian function of system (1), according to (3), takes the form

\[ H(x',y',p',q')=\bar H(x',y',p')+\tilde H(x',y',p',q'), \]

where

\[ \oint \tilde H(x',y',p',q')\,dq'=0 \]

and \(\partial \bar H/\partial x'=\partial \bar H/\partial y'=0\) for \(x'=y'=0\).

The canonical change of variables

\[ x'=X+\partial S/\partial y',\qquad Y=y'+\partial S/\partial X,\qquad p'=P+\partial S/\partial q', \]

\[ Q=q'+\partial S/\partial P \tag{4} \]

with generating function \(V=Xy'+Pq'+S(X,y',P,q')\) brings \(H(x',y',p',q')\) to the form

\[ H=\bar H(X,Y,P)+R_1+R_2+R_3+R_4+R_5, \tag{5} \]

where

\[ R_1=\sum_{\alpha=1}^{n}\frac{\partial S}{\partial q'_{\alpha}}\frac{\partial}{\partial P_{\alpha}} H(X,y',P)+[\tilde H(X,y',P,q')]_N+ \]

\[ +\sum_{\alpha=1}^{m}\left( \frac{\partial S}{\partial y'_{\alpha}}\frac{\partial}{\partial X_{\alpha}} \bar H(X,y',P) -\frac{\partial S}{\partial X_{\alpha}}\frac{\partial}{\partial y'_{\alpha}} \bar H(X,y',P) \right), \]

\[ \begin{gathered} R_2=\overline H(x',y',p')-\overline H(X,y',P)- \\ -\sum_{\alpha=1}^m \frac{\partial S}{\partial y'_\alpha}\, \frac{\partial}{\partial X_\alpha}\overline H(X,y',P) -\sum_{\alpha=1}^n \frac{\partial S}{\partial q'_\alpha}\, \frac{\partial}{\partial P_\alpha}\overline H(X,y',P), \\[4pt] R_3=\overline H(X,y',P)-\overline H(X,Y,P) +\sum_{\alpha=1}^m \frac{\partial S}{\partial X_\alpha}\, \frac{\partial}{\partial y'_\alpha}\overline H(X,y',P), \\[4pt] R_4=\widetilde H(X,y',P,q')-[\widetilde H(X,y',P,q')]_N, \\[4pt] R_5=\widetilde H(x',y',p',q')-\widetilde H(X,y',P,q'), \\[4pt] \widetilde H(x',y',p',q')=\sum_{|k|\ne 0} h_k(x',y',p')e^{i(k,q')}, \\[4pt] [\widetilde H(X,y',P,q')]_N=\sum_{0<|k|<N} h_k(X,y',P)e^{i(k,q')}. \end{gathered} \]

and the variables \(x',y',p',q'\) are expressed in terms of \(X,Y,P,Q\) according to the equalities (4).

Take the function \(S(X,y',P,q')\) in the form
\[ S=S_0+S_X X+S_{y'}y' + {}^1/_2 S_{XX}X^2+S_{Xy'}Xy' + {}^1/_2 S_{y'y'}y'^2, \]
where the function \(S_0=S_0(P,q')\) satisfies the equation
\[ (\omega,\partial S_0/\partial q')+[\widetilde H_0(P,q')]_N=0, \tag{6} \]
the vectors \(S_X=S_X(P,q')\) and \(S_{y'}=S_{y'}(P,q')\) satisfy the system of equations
\[ \left(\omega,\frac{\partial S_X}{\partial q'}\right)+[\widetilde H_X(P,q')]_N = \frac{\partial^2 \overline H}{\partial X\,\partial y'}S_X - \frac{\partial^2 \overline H}{\partial X^2}S_{y'}, \]
\[ \left(\omega,\frac{\partial S_{y'}}{\partial q'}\right)+[\widetilde H_y(P,q')]_N = \frac{\partial^2 \overline H}{\partial y'^2}S_X - \frac{\partial^2 \overline H}{\partial y'\,\partial X}S_{y'}, \tag{7} \]
and the matrices \(S_{XX}=S_{XX}(P,q')\), \(S_{Xy'}=S_{Xy'}(P,q')\), and \(S_{y'y'}=S_{y'y'}(P,q')\) satisfy the system of matrix equations
\[ \left(\omega,\frac{\partial S_{XX}}{\partial q'}\right)+[\widetilde H_{XX}(P,q')]_N = S_{XX}\frac{\partial^2\overline H}{\partial y'\partial X} + \frac{\partial^2\overline H}{\partial X\partial y'}S_{XX} - \]
\[ - S_{Xy'}\frac{\partial^2\overline H}{\partial X^2} - \frac{\partial^2\overline H}{\partial X^2}S'_{Xy'} + \sum_{\alpha=1}^m \left( S_{X_\alpha}\frac{\partial}{\partial y'_\alpha}\frac{\partial^2\overline H}{\partial X^2} - S_{y'_\alpha}\frac{\partial}{\partial X_\alpha}\frac{\partial^2\overline H}{\partial X^2} \right); \]
\[ \left(\omega,\frac{\partial S_{Xy'}}{\partial q'}\right)+[\widetilde H_{Xy'}(P,q')]_N = S_{XX}\frac{\partial^2\overline H}{\partial y'^2} + \frac{\partial^2\overline H}{\partial X\partial y'}S_{Xy'} - \]
\[ - S_{Xy'}\frac{\partial^2\overline H}{\partial X\partial y'} - \frac{\partial^2\overline H}{\partial X^2}S_{y'y'} + \sum_{\alpha=1}^m \left( S_{X_\alpha}\frac{\partial}{\partial y'_\alpha}\frac{\partial^2\overline H}{\partial X\partial y'} - S_{y'_\alpha}\frac{\partial}{\partial X_\alpha}\frac{\partial^2\overline H}{\partial X\partial y'} \right), \tag{8} \]
\[ \left(\omega,\frac{\partial S_{y'y'}}{\partial q'}\right)+[\widetilde H_{y'y'}(P,q')]_N = S'_{Xy'}\frac{\partial^2\overline H}{\partial y'^2} + \frac{\partial^2\overline H}{\partial y'^2}S_{Xy'} - \]
\[ - S_{y'y'}\frac{\partial^2\overline H}{\partial X\partial y'} - \frac{\partial^2\overline H}{\partial y'\partial X}S_{y'y'} + \sum_{\alpha=1}^m \left( S_{X_\alpha}\frac{\partial}{\partial y'_\alpha}\frac{\partial^2\overline H}{\partial y'^2} - S_{y'_\alpha}\frac{\partial}{\partial X_\alpha}\frac{\partial^2\overline H}{\partial y'^2} \right). \]

Here \(\overline H=\overline H(X,y',P)\); \(\omega=\omega(P)=\partial H/\partial P\big|_{X=y'=0}\); the matrices \(\partial^2\overline H/\partial X^2\), \(\partial^2\overline H/\partial X\partial y'\), \(\partial^2\overline H/\partial y'^2\) are taken at \(X=y'=0\); \([\widetilde H_0(P,q')]_N\), \([\widetilde H_X(P,q')]_N\), etc. denote, respectively, the value of the function \([\widetilde H(X,y',P,q')]_N\) and of its partial derivatives at \(X=y'=0\).

Let now in the domain \(G_N^\delta \subset G-\beta\) * the conditions be fulfilled
\[ |(\omega,k)|>\delta |k|^{-(n+1)},\qquad |\lambda_\alpha-i(\omega,k)|>\delta |k|^{-(n+1)}, \]
\[ |\lambda_\alpha+\lambda_\beta-i(\omega,k)|>\delta |k|^{-(n+1)} \]
for all integral values \(k=(k_1,\ldots,k_n)\) such that
\[ |k|=\sum_{j=1}^n |k_j|<N. \]
Then, for sufficiently small \(\delta>0\), in the domain \(P\in G_N^\delta\), \(|\operatorname{Im} q'|<\rho-3\delta\), the system of equations (6), (7), (8) has an analytic solution satisfying in this domain the inequalities
\[ |S_0(P,q')|<M\delta^{-(2n+3)},\qquad |S_X(P,q'),\,S_{y'}(P,q')|<Mh^{-1}\delta^{-(m+1)(n+2)+1}, \]
\[ |S_{XX}(P,q'),\,S_{Xy'}(P,q'),\,S_{y'y'}(P,q')|<Mh^{-2}\delta^{-3m(n+2)+1}. \]

It follows from this that in the domain \(F'\colon |X|<h'\), \(|Y|<h'\), \(P\in G_N^\delta-2\beta\), \(|\operatorname{Im} Q|<\rho'=\rho-3\gamma\), for
\[ N=\frac1\gamma\ln\frac{h^2}{2M} \]
the inequalities
\[ |R_1|<Mh^{-3}h'^3\delta^{-\nu},\qquad |R_2,R_3,R_4,R_5|<M^2h^{-2}\delta^{-2\nu}, \]
are valid, where \(\nu=3m(n+2)\). A corresponding choice of \(\beta>0\) makes it possible to make the measure of the difference \(G\setminus G'\) small together with \(M\). Putting further \(M=\delta^T\), \(h=\delta^\Gamma\), \(h'=h^{1+1/6}\), where \(T=36\nu\), \(\Gamma=14\nu\), we obtain that in the domain \(F'\) inequality (2) is fulfilled; moreover, the partial derivatives of the function \(H_2(X,Y,P,Q)\) with respect to \(X\) and \(Y\) up to and including second order will be small together with \(M\) for \(X=Y=0\). This makes it possible to apply the constructed procedure an unlimited number of times. The convergence of the process is proved analogously to how this is done in work \((^2)\).

In conclusion it is necessary to note that the formulated theorem, with obvious modifications, remains valid also in the case of the canonical system
\[ \dot x=-\partial H/\partial\varphi,\qquad \dot\varphi=\partial H/\partial x \quad (x=(x_1,\ldots,x_m),\ \varphi=(\varphi_1,\ldots,\varphi_m)), \tag{9} \]
\[ \dot p=-\partial H/\partial q,\qquad \dot q=\partial H/\partial p \quad (p=(p_1,\ldots,p_n);\ q=(q_1,\ldots,q_n)) \]
with Hamiltonian function of the form
\[ H=H_0(x,p)+\varepsilon\bigl(H_1(x,p,\varphi)+H_2(x,p,\varphi,q)\bigr), \]
apart from the obvious conditions of analyticity in \(x,p,\varphi,q\) and periodicity in \(q\), satisfying the following conditions:

1. \(\partial H_0/\partial x=\partial H_1/\partial x=\partial H_1/\partial\varphi=0\) for \(x=\varphi=0\).
2. The functional determinants of the matrices \(\partial^2H_0/\partial p^2\), \(\partial^2H_0/\partial x^2\), \(\partial^2H_1/\partial\varphi^2\) do not vanish in the domain \(G\) for \(x=\varphi=0\).
3. The parameter \(\varepsilon\) is sufficiently small.

The finding of conditionally periodic solutions of system (9), close to the solution \(x=\varphi=0\), \(p=p_\omega\) of system (9) with \(H_2(x,p,\varphi,q)\equiv0\), plays an important role in the investigation of the phenomenon of instability in Hamiltonian systems close to integrable ones \((^4)\).

Joint Institute
for Nuclear Research

Received
20 IV 1965

CITED LITERATURE

\(^1\) A. N. Kolmogorov, DAN, 98, No. 4, 527 (1954).
\(^2\) V. I. Arnold, UMN, 18, No. 5, 13 (1963).
\(^3\) M. Born, Lectures on Atomic Mechanics, 1, Kiev, 1934.
\(^4\) V. I. Arnold, DAN, 156, No. 1, 9 (1964).

* The domain \(G-\beta\subset G\) contains those points of the domain \(G\) whose distance to the boundary of the domain \(G\) is greater than \(\beta\) \((\beta>0)\).