Moser Theorem Research Articles

MODULI SPACES OF TOPOLOGICAL CALIBRATIONS, CALABI–YAU, HYPERKÄHLER, G2and SPIN(7) STRUCTURES

This paper focuses on a geometric structure defined by a system of closed exterior differential forms and develops a new approach to deformation problems of geometric structures. We obtain a criterion for unobstructed deformations from a cohomological point of view (Theorem 1.7). Further we show that under a cohomological condition, the moduli space of the geometric structures becomes a smooth manifold of finite dimension (Theorem 1.8). We apply our approach to the geometric structures such as Calabi–Yau, HyperKähler, G2and Spin(7) structures and then obtain a unified construction of smooth moduli spaces of these four geometric structures. We generalize the Moser's stability theorem to provide a direct proof of the local Torelli type theorem in these four geometric structures (Theorem 1.10).

Controllability for a class of area-preserving twist maps

In this paper, we study controllability of two-dimensional integrable twist maps with bounded area-preserving time-dependent (control) perturbations. In contrast to the time-independent perturbation case of the Kolmogorov–Arnold–Moser theorem, there are no invariant sets other than the whole phase space if the perturbation is made a function of time. We give necessary and sufficient conditions for global controllability of these maps.

Relative normal modes for nonlinear Hamiltonian systems

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium. The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighbouring the stable relative equilibrium in question and with any prefixed (spatio-temporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how, in our result, the existence of non-trivial relative periodic orbits requires (generic) conditions on the higher-order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein–Moser theorem, which emphasizes the highly nonlinear character of the relatively periodic dynamical objects.

Diophantine conditions in small divisors and transcendental number theory

We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that $e$ is transcendental over Liouville fields K$(\theta)$ where $\theta $ is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K$(\Theta )$ generated by a family $\Theta$ of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.

Stability of elliptic fixed points of analytic area-preserving mappings under the Bruno condition

We show the stability of an elliptic fixed point of an analytic area-preserving mapping under the Bruno condition, using Moser's Twist-mapping Theorem. We require that there are non-vanishing Birkhoff constants and prove the convergence of the transformation to the Birkhoff normal form if all Birkhoff constants are zero. The convergence theorem is formulated and proved for arbitrary dimensions.

Polarization chaos in nonlinear birefringent resonators

We show numerically and analytically that in a wide class of nonlinear birefringent resonators and periodic structures, a special type of polarization chaos related to the Kolmogorov–Arnold–Moser (KAM) theorem takes place. In this case regions of regular behavior on the Poincaré sphere coexist with regions of chaotic one. The effect presumably can be observed in fiber lasers or in periodically twisted birefringent optical fibers.

Some properties of locally conformal symplectic structures

We show that the d_{\omega} -cohomology is isomorphic to a conformally invariant usual de Rham cohomology of an appropriate cover. We also prove a Moser theorem for locally conformal symplectic (lcs) forms. We point out a connection between lcs geometry and contact geometry. Finally, we show the connections between first kind, second kind, essential, inessential, local, and global conformal symplectic structures through several invariants.

Transition from order to chaos in a floppy molecule: LiNC/LiCN

The transition from order to chaos in molecular systems of low dimensionality is best understood with the aid of the rigorous Kolmogorov–Arnold–Moser theorem. However, in the late 1970s, some authors proposed a less abstract criterion, which is based on extreme sensitivity to initial conditions exhibited by chaotic trajectories and that has a straightforward connection to features of the potential energy surface. An extension of this criterion is proposed and applied to the study of the floppy system LiNC/LiCN. The correspondence with the quantum results is also discussed.

Perturbations of Lower Dimensional Tori for Hamiltonian Systems

This paper provides a generalized Kolmogorov–Arnold–Moser theorem for lower dimensional tori in Hamiltonian systems, which applies to multiple normal frequency case. The proof is based on Newton's iteration method and a generalized version of small divisor conditions.

A simple proof of Moser's theorem

This article gives a simple proof of a result of Moser, which says that, for any rational number r between 2 and 3, there exists a planar graph G whose circular chromatic number is equal to r. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 19–26, 1999

A simple proof of Moser's theorem

A simple proof of Moser's theorem

A version of Thirring’s approach to the Kolmogorov–Arnold–Moser theorem for quadratic Hamiltonians with degenerate twist

We give a proof of the Kolmogorov–Arnold–Moser (KAM) theorem on the existence of invariant tori for weakly perturbed Hamiltonian systems, based on Thirring’s approach for Hamiltonians that are quadratic in the action variables. The main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic Hamiltonians. We show that Thirring’s proof for nondegenerate Hamiltonians can be adapted to Hamiltonians with degenerate twist. This case, in fact, drastically simplifies Thirring’s proof.

Quasi Linear Flows on Tori:�Regularity of their Linearization

Under suitable conditions a flow on a torus $C^{(p)}$--close, with $p$ large enough, to a quasi periodic diophantine rotation is shown to be conjugated to the quasi periodic rotation by a map that is analytic in the perturbation size. This result is parallel to Moser's theorem stating conjugability in class $C^{(p')}$ for some $p'<p$. The extra conditions restrict the class of perturbations that are allowed.

Can a Local Repulsive Potential Trap an Electron?

We study the classical dynamics of a charged particle in two dimensions, under the influence of a perpendicular magnetic and an in-plane electric field. We prove the surprising fact that there is a finite region in phase space that corresponds to the otherwise drifting particle being trapped by a local repulsive potential. Our result is a direct consequence of Kolmogorov-Arnold-Moser theory and, in particular, of Moser's theorem. We illustrate it by numerical phase portraits and by an analytic approximation to invariant curves.

EXISTENCE OF INVARIANT CURVES FOR MAPS CLOSE TO DEGENERATE MAPS, AND A SOLUTION OF THE FERMI-ULAM PROBLEM

The Ulam model is studied in this paper: a small elastic ball moves vertically between two infinitely heavy horizontal walls, each of which moves in the vertical direction according to a periodic law. It is proved that the velocity of the ball is always bounded. The proof is based on a generalization of Moser's theorem on the existence of invariant curves under an area preserving mapping of an annulus. Bibliography: 16 titles.

Q spoiling and directionality in deformed ring cavities

A ray-optics model is developed to describe the spoiling of the high-Q (whispering gallery) modes of ring-shaped cavities as they are deformed from perfect circularity. A sharp threshold is found for the onset of Q spoiling as predicted by the Kolmogorov-Arnol'd-Moser (KAM) theorem of nonlinear dynamics. Beyond the critical deformation b(c), Q ~ (b - b(c))(-alpha), alpha asymptotically equal to 2.4-2.6. The escaping light emerges in certain specific directions, which may be predicted.

Hamiltonian truncation of the shallow water equation

In this Letter, we describe a truncation of the Eulerian description of the shallow water equation of climate modeling to a finite-dimensional Hamiltonian system. The technique is to use an isomorphism from a semidirect product Poisson manifold to a direct product of Poisson manifolds, both of whose components are truncatable to finite-dimensional Poisson manifolds, based on Moser's theorem on volume elements.

Integrals, invariant manifolds, and degeneracy for central force problems in R n

Since the notion of angular momentum is defined in any dimension by using the exterior product in Rn, one would guess that central force problems in any dimension are completely integrable, as it is known for n=2 or 3. This is proved explicitly in this paper, by constructing n first integrals independent and involution: the energy and some combinations of the angular momentum components. It is shown that these problems are always reduced to a two-dimensional plane, and the invariant manifolds are topologically, the Cartesian product of those of the reduced problem times n−2 circle factors. It is also proved that for n≳2 these problems are degenerate in the sense that their Hamiltonians do not satisfy one of the hypotheses of the Kolmogorov–Arnold–Moser theorem for persistence of invariant tori, when one considers small perturbations.

On weakly harmonic maps and Noether harmonic maps from a Riemann surface into a Riemannian manifold

I want to present some results on the regularity for harmonic maps between a surface of dimension two and a Riemannian manifold. First of all, we recall what harmonic maps between Riemannian manifolds are. Let (M, g) and (N , g) be two Riemannian manifolds of dimension m and n respectively, and assume that N is compact and is isometrically embedded in some Euclidean space R (which is always possible thanks to the Nash–Moser theorem). We introduce the Dirichlet functional on the set of maps between M and N . To do this we define the energy density of a map u from M into N at any point x of M by e(u)(x) = 1 2hij [u(x)]g (x)uαu j β

Kolmogorov stability, the impossibility of Fermi acceleration and the existence of periodic solutions in some Hamiltonian-type systems

In certain special systems, pertaining to various branches of mechanics and physics, there is no explicitly occurring small parameter, but a suitable change of variables will artificially introduce a small parameter into the equations of motion. One can then use the Kolmogorov-Arnol'd-Moser Theorem to prove the existence of invariant (Kolmogorov) tori, which, moreover, fill the whole of phase space except for a set of finite measure (the measure of the whole phase space is infinite). Bounds are obtained for the deformation of invariant tori and the relative measure of the Kolmogorov set. It follows from Poincaré's Geometric Theorem that in all problems under consideration there exist infinitely many periodic solutions. The results of numerical experiments in connection with Ulam's problem receive their first comprehensive, rigorous justification, and analogous propositions are derived for other situations. The case of pendulum-type systems is then considered in this context by treating the reciprocal of the momentum as the small parameter. This device was first used in connection with the equations of mechanics (when momentum = angular velocity) to determine the asymptotic behaviour of fast rotations [1]. In the case of Ulam's problem a small parameter reciprocal to the particle velocity was introduced in [2]. This paper generalizes and develops results of [3] which follows from Theorem 1 as a special case.