V.I. Arnol'd, A. Avez: "Ergodic problems of classical mechanics", Benjamin (1968).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....for hyperbolic geometry; see [48] for details. With the canonical momenta p a = L= q = mg ab q , the classical Hamiltonian reads H(p; q) p a g p b , where g ac g cb = b . This dynamical system has become a central object in the development of ergodic theory ever since [49]. Hadamard [47] proved already that all trajectories in this system are unstable and that neighbouring trajectories diverge in time at an exponential rate, the most striking property of deterministic chaos. Thus Hadamard s billiard possesses the properties of an Anosov system [49] The quantum ....

....theory ever since [49] Hadamard [47] proved already that all trajectories in this system are unstable and that neighbouring trajectories diverge in time at an exponential rate, the most striking property of deterministic chaos. Thus Hadamard s billiard possesses the properties of an Anosov system [49]. The quantum mechanics of the Hadamard Gutzwiller model is determined [46,12,48] by the Schr odinger equation (1) with 2m LB , where LB = 1=2 a G ab b is the Laplace Beltrami operator on the D dimensional hyperbolic manifold with G = G(q) det(g ab (q) In addition, the wave ....

V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics (Addison-Wesley 1968). 21

....manifolds Consider the torus T = 1 2 g as a symplectic manifold with the two form dq A dp. Then any M SL(2, Z) defines a (discrete) symplectic dynamics on T in the obvious way. We are interested in the case where M is hyperbolic: the corresponding dynamical system is then an Anosov system [AA]. The stable and unstable manifolds of any point x T are obtained by wrapping the lines with slopes s passing through x around the torus. We present here some properties of these manifolds that we will need in subsequent sections. A simple example we will use for numerical illustrations is the so ....

....manifolds of any point x T are obtained by wrapping the lines with slopes s passing through x around the torus. We present here some properties of these manifolds that we will need in subsequent sections. A simple example we will use for numerical illustrations is the so called Arnold s cat map [AA] MArnold = 2 1) 1 1 (21) Its L yapounov coefficient is A0 = log (3 ) 0.9624. The stable and unstable manifolds of the fixed point x = 0 are depicted in Figure 4. 0.5 0 pt 0.5 0 0.5 Figure 4: The stable and unstable axes through 0 of the map MArhold wrap around the torus at infinity. ....

V.I. Arnold and A. Avez, Ergodic Problems in Classical Mechanics, Benjamin, New York, 1968

....for FPU s observations, although the relation between the FPU chain and its infinite dimensional limits has never been completely understood. Another, possibly correct explanation for the quasiperiodic behaviour of the FPU system, is based on the Kolmogorov Arnol d Moser theorem. As is well known [1], the solutions of an n degree of freedom Liouville integrable Hamiltonian system are constrained to move on n dimensional tori and are not at all ergodic but periodic and quasiperiodic. The KAM theorem states that most invariant tori of such an integrable system persist under small Hamiltonian ....

V. I. Arnol d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

....theory [130] although not as complete for the nonself adjoint Zakharov Shabat operator (3.4) as for the Hill s operator [161] shows that 1 D NLS (3.1) under periodic boundary conditions is a completely integrable Hamiltonian system. Its integration is accomplished through Louiville s method [5, 4], as realized by an Abel Jacobi transformation and theta functions. This procedure amounts to a transformation from q(x) to action angle variables [99, 98] a beautiful procedure which is most easily described for soliton equations in the case of the Toda lattice [69, 71, 43] Generically, the ....

V. I. Arnold and A. Avez. Ergodic problems of classical mechanics. W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez.

....rotation number of M(t; t 0 ) exists and is equal to ae: The proof of both claims is elementary. 2 2 Laplace construction of C We now further specify the matrix C = C(t) using a construction to study the well known Laplace problem on the mean motion of the perihelia of the planets, compare [1, 3]. We describe this specification in a number of steps. A 1 A 3 A 2 g 2 g 3 g 1 Figure 1: The triangle with sides jA 1 j; jA 2 j and jA 3 j and corresponding opposite angles jfl 1 j; jfl 2 j and jfl 3 j. 1. One ingredient consists of three complex numbers A 1 ; A 2 and A 3 ; such ....

Arnol 0 d, V.I. and Avez, A.: Ergodic Problems of Classical Mechanics. Benjamin 1968.

....theory [130] although not as complete for the nonself adjoint Zakharov Shabat operator (3.4) as for the Hill s operator [161] shows that 1 D NLS (3.1) under periodic boundary conditions is a completely integrable Hamiltonian system. Its integration is accomplished through Louiville s method [5, 4], as realized by an Abel Jacobi transformation and theta functions. This procedure amounts to a transformation from q(x) to action angle variables [99, 98] a beautiful procedure which is most easily described for soliton equations in the case of the Toda lattice [69, 71, 43] Generically, the ....

V. I. Arnold and A. Avez. Ergodic problems of classical mechanics. W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez.

....dynamical systems in order to convince to the reader that natural actions of linear free groups can give rise to very complicated dynamics. The Anosov map on the real two dimensional torus T 2 = R 2 =Z 2 is de ned by the action of the matrix = 1 1 1 2 ; on R 2 , mod 1) see [7, 6, 26]. As every body knows the discrete dynamical system de ned by the action of the above map on T 2 is chaotic: all the set T 2 is an hyperbolic attractor, it has an in nite number of periodic orbits, etc. It is possible to look at the Anosov map from another point of view. This map is ....

V.I. Arnold, A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York, 1968.

....Since the proposed coupling is uni directional, 2) has a skew product structure that allows us to consider the x dynamics in isolation. In the absence of a message, x n 1 = Ax n mod 1 : 3) When A is hyperbolic with Det A = 1, 3) is a hyperbolic toral automorphism, or Anosov system [31, 32]. When N = 2 a famous example is Arnold s cat map [31] Traditionally written with A = 1 1 1 2 ) in upper companion form the matrix becomes 3 1 1 0 . The resulting system remains a hyperbolic toral automorphism with the same eigenvalues and hence the same asymptotic behavior as the cat ....

....(2) has a skew product structure that allows us to consider the x dynamics in isolation. In the absence of a message, x n 1 = Ax n mod 1 : 3) When A is hyperbolic with Det A = 1, 3) is a hyperbolic toral automorphism, or Anosov system [31, 32] When N = 2 a famous example is Arnold s cat map [31]. Traditionally written with A = 1 1 1 2 ) in upper companion form the matrix becomes 3 1 1 0 . The resulting system remains a hyperbolic toral automorphism with the same eigenvalues and hence the same asymptotic behavior as the cat map. In [33] the authors studied systems similar to (2) ....

V.I. Arnold and A. Avez. Ergodic Problems of Classical Mechanics. Benjamin, New York, 1968.

....is not a well established notion, but rather 3 chaoticity encompasses many mathematical definitions such as mixing, Anosov systems, K systems and others) We will use some of these definitions in this part of the work, assuming the reader is well familiar with them. Standard references include [AA, CFS, Ma, W]. 1.1 Billiards We have said that the vast majority of the systems in which physicists are interested cannot be integrated. As a matter of fact, they are hard to study from the point of view of ergodic theory, too. Therefore mathematicians are forced to consider models, abstract simplified ....

V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, W. A. Benjamin, New York, 1968.

....act ergodically and with zero entropy by continuous homomorphisms on any compact abelian group. Kushnirenko showed that Z cannot act with infinite entropy by smooth maps of a smooth manifold. The relevant definitions and a proof of this can be found in sections 1 and 12 of Arnold and Avez s book [2]. An interesting example of group constraints dates back to a famous open algebraic problem first posed by Lehmer [31] in 1933. It is well known (see, for example Walters [64, Section 0.8] that any continuous group automorphism of the n dimensional additive torus group S n (viewed as the ....

V. I. Arnold & A. Avez, Ergodic Problems in Classical Mechanics, AddisonWesley, Reading, Mass., (1989).

....r.schack rhbnc.ac.uk y E mail: caves tangelo.phys.unm.edu 1 quantum baker s map. Shifts on in nite quantum spin chains in the context of quantum chaos have been discussed in [11] A related symbolic description of the quantum baker s map is given in [12] The classical baker s transformation [13], which maps the unit square 0 q; p 1 onto itself, has a simple description in terms of its symbolic dynamics [4] Each point in phase space is represented by a symbolic string s = s 2 s 1 s 0 :s 1 s 2 ; 1) where s k = 0 or 1. The string s is identi ed with a point (q; p) in the ....

Arnold, V.I., Avez, A. (1968): Ergodic Problems of Classical Mechanics. Benjamin, New York

....s. A straightforward calculation shows that M bijectively maps the fundamental domain onto itself (Fig. 5) Moreover, the dynamics locally preserves the area of volume elements such that an initially uniform density stays uniform at all times. 2 Since the Lorentz gas has a mixing dynamics (cf. [34, 57] for details) every smooth initial density will asymptotically approach this uniform density. The motion of a particle in the field free Lorentz gas can be traced backward by inverting its velocity at any given time. At a collision this corresponds to the action b n 1 7 Gammab n 1 and n 1 7 ....

V.I. Arnold and A. Avez, Ergodic problems of classical mechanics (Benjamin, New York, 1968).

.... state respectively, see e.g. 16, 17, 33, 41, 62] Yet, in analogy with the classical case, we prefer to retain our terminology since properties (iii) and (iv) are the natural generalizations to quantum (i.e. non commutative) cases of analogous properties considered in commutative cases, see e.g. [4, 31]. Noticing that we have the chain of implications (iv) iii) ii) for the above properties ( 41] Theorem 2.1) we focus our attention only on properties (i) and (iv) Finally, we remark that, for asymptotically Abelian QDS (A; ff t ; property (iv) is equivalent to lim t 1 (Aff t B) ....

Arnold V. I., Avez A. Ergodic problems of classical mechanics, Benjamin, New York, 1968.

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V.I. Arnol'd, A. Avez: "Ergodic problems of classical mechanics", Benjamin (1968).

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Arnold, A., Avez, A.: Ergodic Problems of Classical Mechanics. New York: Benjamin, 1968

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Arnold, V., Avez, A.: Ergodic problems of classical mechanics, Benjamin, 1966.

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V.I. Arnold and A. Avez. Ergodic problems of classical mechanics. Benjamin, New York, 1967.

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V.I. Arnold and A. Avez. Ergodic problems of classical mechanics. Benjamin, New York, 1967.

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V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York, 1968.

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V. Arnold and A. Avec, Ergodic problems of classical mechanics. Benjamin, New York. 1968.

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V. Arnold and A. Avec, Ergodic problems of classical mechanics. Benjamin, New York. 1968.

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V. Arnold and A. Avec, Ergodic problems of classical mechanics. Benjamin, New York. 1968.

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Arnold, V. I., and Avez, A. Ergodic Problems of Classical Mechanics. Benjamin, New York, 1968.

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V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York, 1968.

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V.I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, (AddisonWesley, Redwood City 1989)

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