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1.
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville—Arnold
integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytical method for the investigation
of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically
the structure of quasiperiodic solutions of the Hamiltonian system under consideration. We also consider the problem of existence
of adiabatic invariants associated with a slowly perturbed Hamiltonian system.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 11, pp. 1513–1528, November, 1999. 相似文献
2.
We study the problem of perturbations of quasiperiodic motions in the class of locally Hamiltonian systems. By using methods
of the KAM-theory, we prove a theorem on the existence of invariant tori of locally Hamiltonian systems close to conditionally
integrable systems. On the basis of this theorem, we investigate the bifurcation of a Cantor set of invariant tori in the
case where a Liouville-integrable system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic
structure of the phase space is deformed.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 71–98, January, 2007. 相似文献
3.
B. M. Gurevich 《Theoretical and Mathematical Physics》1992,90(3):289-312
The Liouville operator for an infinite-particle Hamiltonian dynamics corresponding to interaction potentialU is used to introduce the concept of a locally weakly invariant measure on the phase space and to show that if a Gibbs measure with potential of general form is locally weakly invariant then its Hamiltonian is asymptotically an additive integral of the motion of the particles with the interactionU.Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 424–459, March, 1992. 相似文献
4.
We study the class of pseudo-norms on the space of smooth functions on a closed symplectic manifold, which are invariant under
the action of the group of Hamiltonian diffeomorphisms. Our main result shows that any such pseudo-norm that is continuous
with respect to the C
∞-topology, is dominated from above by the L
∞-norm. As a corollary, we obtain that any bi-invariant Finsler pseudo-metric on the group of Hamiltonian diffeomorphisms that
is generated by an invariant pseudonorm that satisfies the aforementioned continuity assumption, is either identically zero
or equivalent to Hofer’s metric. 相似文献
5.
In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus,numerical solutions are quasi-periodic with a diophantine frequency vector of time step size dependence. These results generalize Shang's previous ones(1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov. 相似文献
6.
We investigate the influence of small deformations of a symplectic structure and perturbations of the Hamiltonian on the behavior of a completely integrable Hamiltonian system. We show that a Whitney-smooth family of coisotropic invariant tori of the perturbed system emerges in the neighborhood of a certain submanifold of the phase space. 相似文献
7.
We give a fairly simple geometric proof that an equilibrium point of a Hamiltonian system of two degrees of freedom is Liapunov
stable in a degenerate case. That is the 1: −1 resonance case where the linearized system has double pure imaginary eigenvalues
±iω, ω ≠ 0 and the Hamiltonian is indefinite. The linear system is weakly unstable, but if a particular coefficient in the normalized
Hamiltonian is of the correct sign then Moser’s invariant curve theorem can be applied to show that the equilibrium point
is encased in invariant tori and thus it is stable. 相似文献
8.
In this paper, we prove the persistence of hyperbolic lower dimensional invariant tori for Gevrey-smooth perturbations of partially integrable Hamiltonian systems under Riissmann's nondegeneracy condition by an improved KAM iteration, and the persisting invariant tori are Gevrey smooth, with the same Gevrey index as the Hamiltonian. 相似文献
9.
Tan Su 《Regular and Chaotic Dynamics》2012,17(1):54-62
Let the adiabatic invariant of action variable in a slow-fast Hamiltonian system with two degrees of freedom have limits along
the trajectories as time tends to plus and minus infinity. The difference of these two limits is exponentially small in analytic
systems. An isoenergetic reduction and canonical transformations are applied to transform the slow-fast system to form of
a system depending on a slowly varying parameter in a complexified phase space. On the basis of this method an estimate for
the accuracy of conservation of adiabatic invariant is given. 相似文献
10.
We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω 0 is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω 0 has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω 0. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point. 相似文献
11.
Ya. A. Prykarpats’kyi 《Ukrainian Mathematical Journal》2006,58(5):763-778
We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic
manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of
invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and
others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog
of a Mather-type β-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly
exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we
establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic
invariants of slowly perturbed integrable Hamiltonian systems.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 675–691, May, 2006. 相似文献
12.
Alberto Abbondandolo Alessio Figalli 《Journal of Fixed Point Theory and Applications》2008,3(1):95-120
We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the
cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic
Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of
an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure
with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of
Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector.
Dedicated to Vladimir Igorevich Arnold 相似文献
13.
A. P. Krishchenko 《Computational Mathematics and Modeling》2011,22(4):361-373
A class of problems that may be characterized as localization problems are becoming increasingly popular in qualitative theory
of differential equations [1–15]. The specific formulations differ, but geometrically all search for phase space subsets with
desired properties, e.g., contain certain solutions of the system of differential equations. Such problems include construction
of positive invariant sets that contain certain separatrices of the Lorenz system [1], analysis of asymptotic behavior of
solutions of the Lorenz system and determination of sets that contain the Lorenz attractor [2–5, 14], as well as determination
of sets containing all periodic trajectories [6–13], separatrices, and other trajectories [10, 11]. Such sets may be naturally
called localizing sets and it is obviously interesting to study methods and results that produce exact or nearly exact localizing
sets for each phase space structure. In this article we focus on localization of the invariant compact sets in the phase space
of a differential equation system, specifically the problem of finding phase space subsets that contain all the invariant
compacta of the system. Invariant compact sets are equilibria, periodic trajectories, separatrices, limit cycles, invariant
tori, and other sets and their finite unions. These sets and their properties largely determine the phase space structure
and the qualitative behavior of solutions of the differential equation system. 相似文献
14.
S. A. Frolov 《Theoretical and Mathematical Physics》1997,113(1):1289-1298
It is shown that the physical phase space of the γ-deformed Hamiltonian lattice in the Yang-Mills theory coincides as a Poisson
manifold with the moduli space of flat connections on a Riemann surface with L−V+1 handles and, therefore, with the physical
phase space of the corresponding (2+1)-dimensional Chern-Simons model. Here, L and V are, respectively, the total number of
links and vertices of the lattice. The deformation parameter γ is identified with 2π/k, where k is an integer appearing in
the Chern-Simons action.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 100–111, October, 1997. 相似文献
15.
本文研究了不定度规空间空间中的无穷维Hamilton算子.利用Plus算子存在极大不变子空间的性质,获得了无穷维Hamilton算子在Krein空间中存在极大确定不变子空间的充分条件. 相似文献
16.
Robert S. MacKay 《Regular and Chaotic Dynamics》2018,23(6):797-802
A method is presented to establish regions of phase space for 3D vector fields through which pass no co-oriented invariant 2D submanifolds transverse to a given oriented 1D foliation. Refinements are given for the cases of volume-preserving or Cartan–Arnol’d Hamiltonian flows and for boundaryless submanifolds. 相似文献
17.
《Communications in Nonlinear Science & Numerical Simulation》2014,19(8):2660-2675
The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem. 相似文献
18.
KAM theory in configuration space 总被引:6,自引:0,他引:6
A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies
is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase
space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely
many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The
Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser.
This research has been supported by the Nuffields Foundation under grant SCI/180/173/G and by the Stiftung Volkswagenwerk. 相似文献
19.
Given a Hamiltonian system, one can represent it using a symplectic map. This symplectic map is specified by a set of homogeneous polynomials which are uniquely determined by the Hamiltonian. In this paper, we construct an invariant norm in the space of homogeneous polynomials of a given degree. This norm is a function of parameters characterizing the original Hamiltonian system. Such a norm has several potential applications. 相似文献
20.
Summary The real-valued Maxwell-Bloch equations on ℝ3 are investigated as a Hamiltonian dynamical system obtained by applying an S1 reduction to an invariant subsystem of a dynamical system on ℂ3. These equations on ℝ3 are bi-Hamiltonian and possess several inequivalent Lie-Poisson structures parametrized by classes of orbits in the group
SL(2, ℝ). Each Lie-Poisson structure possesses an associated Casimir function. When reduced to level sets of these functions,
the motion takes various symplectic forms, from that of the pendulum to that of the Duffing oscillator. The values of the
geometric (Hannay-Berry) phases obtained in reconstructing the solutions are found to depend upon the choice of Casimir function,
that is, upon the parametrization of the reduced symplectic space. 相似文献