×

zbMATH — the first resource for mathematics

On elliptic lower dimensional tori in Hamiltonian systems. (English) Zbl 0662.58037
This paper presents a perturbation theory of KAM-type for finite dimensional, elliptic invariant tori in finite or infinite dimensional Hamiltonian systems. In the model problem, the unperturbed integrable Hamiltonian is \[ N=\sum^{n}_{i=1}\omega_ iy_ i+\sum^{m}_{j=1}\Omega_ j(u^ 2_ j+v^ 2_ j) \] where \(2\leq n<\infty\) and \(1\leq m\leq \infty\), and the underlying phase space is \({\mathbb{T}}^ n\times {\mathbb{R}}^ n\times {\mathbb{R}}^ m\times {\mathbb{R}}^ m\) with coordinates (x,y,u,v). The \(\omega_ i\), \(\Omega_ j\) are real frequencies, and the equations of motion are \(\dot x=\omega\), \(\dot y=0\), \(\dot u=\Omega v\), \(\dot v=-\Omega u\) in usual vector notation. In this setting the frequencies \(\omega\) on the torus \({\mathbb{T}}^ n\) are considered as parameters, while the frequencies \(\Omega\) of the elliptic fixed point \((u,v)=(0,0)\) are usually functions of \(\omega\).
The result is that such a configuration persists under small real analytic perturbations of the Hamiltonian N. The perturbation has to be small in a special weighted norm, and the dependence of \(\Omega\) on \(\omega\) must not be too degenerate to avoid certain low order resonances.
Reviewer: J.Pöschel

MSC:
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] [Arn] Arnold, V.I.:Proof of a theorem by A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Usp. Math. Nauk18, 13-40 (1963); Russ. Math. Surv.18, 9-36 (1963)
[2] [Brj] Brjuno, A.D.:Analytic form of differential equations. Trans. Mosc. Math. Soc.25, 131-288 (1971)
[3] [CC] Celletti, A., Chierchia, L.:Construction of analytic KAM surfaces and effective stability bounds. Commun. Math. Phys.118, 119-161 (1988) · Zbl 0657.58032
[4] [DRV] Dodson, M.M., Rynne, B.P., Vickers, J.A.G.:Metric diophantine approximation and Hausdorff dimensions on manifolds. Math. Proc. Camb. Philos. Soc. (to appear)
[5] [Eli] Eliasson, L.H.:Perturbations of stable invariant tori. Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser.15, 115-147 (1988) · Zbl 0685.58024
[6] [ESW] Fröhlich, J., Spencer, T., Wayne, C.E.:Localization in disordered, nonlinear dynamical systems. J. Stat. Phys.42, 247-274 (1986) · Zbl 0629.60105
[7] [Gra] Graff, S.M.:On the continuation of hyperbolic invariant tori for hamiltonian systems. Jour. Differ. Equations.15, 1-69 (1974) · Zbl 0268.34051
[8] [Kol] Kolmogrov, A.N.:On the conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk. SSSR98, 525-530 (1954)
[9] [Koz] Kozlov, S.M.:Reducibility of quasi-periodic differential operators and averaging. Trans. Mosc. Math. Soc.46, 101-126 (1984) · Zbl 0566.35036
[10] [Kuk] Kuksin, S.B.:Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum. Funkts. Anal. Prilozh.21, 22-37 (1987); Funct. Anal. Appl.21, 192-205 (1988) · Zbl 0631.34069
[11] [Mel-1] Melnikov, V.K.:On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function. Sov. Math., Dokl.6, 1592-1596 (1965) · Zbl 0143.11801
[12] [Mel-2] Melnikov, V.K.:A family of conditionally periodic solutions of a Hamiltonian system. Sov. Math. Dokl.9, 882-886 (1968) · Zbl 0185.17101
[13] [Mos-1] Moser, J.:On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gött., Math.-Phys. Kl. 1-20 (1962) · Zbl 0107.29301
[14] [Mos-2] Moser, J.:Convergent series expansions for quasi-periodic motions. Math. Ann.169, 136-176 (1967) · Zbl 0149.29903
[15] [Mos-3] Moser, J.:A stability theorem for minimal foliations on a torus. Ergodic Theory Dyn. Syst.8, 251-281 (1988) · Zbl 0632.57018
[16] [Nik] Nikolenko, N.V.:The method of Poincaré normal forms in problems of integrability of equations of evolution type. Russ. Math. Surv.41, 63-114 (1986) · Zbl 0632.35026
[17] [Pös-1] Pöschel, J.:Integrability of Hamiltonian systems on Cantor sets. Commun. Pure Appl. Math.35, 653-695 (1982) · Zbl 0542.58015
[18] [Pös-2] Pöschel, J.:On invariant manifolds of complex analytic mappings near fixed points. Expo. Math.4, 97-109 (1986) · Zbl 0597.32011
[19] [Pös-3] Pöschel, J.:A general infinite dimensional KAM-theorem. In:Proceedings of the IX International Congress on Mathematical Physics, Swansea (1988) · Zbl 0729.70019
[20] [Pös-4] Pöschel, J.:Small divisors with spatial structure. Bonn University, 1989 (preprint)
[21] [PT] Pöschel, J., Trubowitz, E.:Inverse Spectral Theory. Boston: Academic Press (1987) · Zbl 0623.34001
[22] [Rüs-1] Rüssmann, H.:On the one-dimensional Schrödinger equation with a quasi-periodic potential. Annals of the New York Acad. Sci.357, 90-107 (1980)
[23] [Rüs-2] Rüssmann, H.: Talk held at the ETH Zürich February 1986
[24] [Sie-1] Siegel, C.L.:Iteration of analytic functions. Ann. Math.43, 607-612 (1942) · Zbl 0061.14904
[25] [Sie-2] Siegel, C.L.:Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. 21-30 (1952) · Zbl 0047.32901
[26] [SM] Siegel, C.L., Moser, J.:Lectures on Celestial Mechanics. Berlin-Heidelberg-New York: Springer, (1971) · Zbl 0312.70017
[27] [Spr] Sprind?uk, V.G.:Theory of Diophantine Approximations. New York: Wiley (1979)
[28] [St] Stein, E.:Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton University Press (1970) · Zbl 0207.13501
[29] [SZ] Salamon, D., Zehnder, E.:KAM-theory in configuration space. Comment. Math. Helv.64, 84-132 (1989) · Zbl 0682.58014
[30] [Vit] Vittot, M.:Théorie classique des perturbations et grand nombre de degres de liberté. Thèse de doctorat de l’universitè de Provence 1985
[31] [VB] Vittot, M., Bellissard, J.:Invariant tori for an infinite lattice of coupled classical rotators. CPT-Marseille, 1985 (preprint)
[32] [War] Ware, B.:Infinite-dimensional versions of two theorems by Carl Siegel. Bull. Am. Math. Soc.82, 613-615 (1976) · Zbl 0344.34029
[33] [Way] Wayne, C.E.:Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Preprint No 88027, Penn. State University, 1988
[34] [Whit] Whitney, H.:Analytic extensions of differentiable functions defined in closed sets. Trans. A.M.S.36, 63-89 (1934) · Zbl 0008.24902
[35] [Zeh-1] Zehnder, E.:Generalized implicit function theorems with applications to some small divisor problems, I and II. Commun. Pure Appl. Math.28, 91-140 (1975); 49-111 (1976) · Zbl 0309.58006
[36] [Zeh-2] Zehnder, E.:Siegel’s linearization theorem in infinite dimensions. Manus. math.23, 363-371 (1978) · Zbl 0374.47037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.