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On the conservation of hyperbolic invariant tori for Hamiltonian systems. (English) Zbl 0257.34048

A result is discussed which is a generalization of a theorem by Kolmogorov and Arnold an the perturbation of invariant tori for real analytic Hamiltonian systems. Consider the real analytic Hamiltonian \(\mathcal H(x,y,p,q) = F(y,p,q)\) with conjugate variables \((x,y)\), \((p,q)\) where \(x\in T^n\), \(y\in\mathbb R^n\), \(p\in\mathbb R^n\) and \(q\in\mathbb R^m\). Assume \(F_p(y, 0, 0) = 0\), \(F_q(y, 0, 0) = 0\), and that the eigenvalues of \(J^{-1}F_{vv}(y, 0, 0)\), \(v = (p,q)\), \(J = \left(\begin{smallmatrix} 0 & I\\ -I & 0\end{smallmatrix}\right)\), do not lie an the imaginary axis. The associated system of differential equations possesses a family of invariant tori \(y = \overset\circ y\), \(p = 0\), \(q = 0\). In addition, there exist two unique \(n+m\) dimensional real analytic manifolds \(M_+(\overset\circ y)\) and \(M_-(\overset\circ y)\) upon which solutions, respectively, leave and approach each invariant torus \(y=\overset\circ y\), \(p=0\), \(q=0\) exponentially and tangentially. The following statement is proved.
Theorem: Let \(\overset\circ y\) be chosen such that the components of \(\omega = F_y(\overset\circ y,0,0)\) are rationally independent and satisfy the irrationality condition \(| (j,\omega)|\geq K| j|^{-(n+1)}\) for some \(K\) and all integer valued vectors \(j\). Suppose \(F_{yy}^{-1}(y,0,0)\) exists and is real analytic. Then, the invariant torus \(y=\overset\circ y\), \(p=0\), \(q=0\) may be continued analytically under any sufficiently small real analytic perturbation \(H(x,y,p,q)\) of the Hamiltonian \(\mathcal H(x,y,p,q)\). The continued torus is represented by an embedding \(\mathcal U: \xi\to (x,y, p,q) \in T^n\times \mathbb R^{n+2m}\). Moreover the \(n+m\)-dimensional manifolds \(M_+(\overset\circ y)\) and \(M_-(\overset\circ y)\) also persist.
The existence of the continued torus is proved by using a Newton iteration procedure. No restriction is placed upon the multiplicity of the eigenvalues of \(J^{-1} F_{vv}(y,0,0)\). The results, concerning \(M_+(\overset\circ y)\) and \(M_-(\overset\circ y)\) are an extension of work done by A. Kelley [J. Differ. Equations 3, 546–570 (1967; Zbl 0173.11001)]. In addition, the Hamiltonian character of the problem implies that \(M_+(\overset\circ y)\) and \(M_-(\overset\circ y)\) are Lagrangian manifolds. The theorem is applicable to the study of an equilibrium of a Hamiltonian system at which the purely imaginary eigenvalues of the coefficient matrix of the linearized system occur in distinct pairs.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H08 Nearly integrable Hamiltonian systems, KAM theory
70H09 Perturbation theories for problems in Hamiltonian and Lagrangian mechanics
34E10 Perturbations, asymptotics of solutions to ordinary differential equations

Citations:

Zbl 0173.11001
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References:

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