KAM theorems for the product of two involutions of different types

The properties of quasiperiodic motions in Hamiltonian, volume preserving, and reversible systems are summarized. KAM theorems concerning lower-dimensional invariant tori are announced for G-reversible mappings A such that the fixed point manifolds Fix(G) and Fix(AG) of the reversing involutions G and AG are of different dimensions. The case where the manifold Fix(G) itself consists of several connected components of different dimensions is also briefly discussed.     

Lower-dimensional tori in reversible systems

On a (2n+d)-dimensional manifold M consider a vector field V reversible with respect to an involution G whose fixed point manifold is of dimension n+d. It is conjectured that generically for each 0相似文献   

Whitney smooth families of invariant tori within the reversible context 2 of KAM theory

We prove a general theorem on the persistence of Whitney C ^∞-smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim FixG < (codim T)/2, where FixG is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H. W.Broer, M.-C.Ciocci, H.Hanßmann, and A.Vanderbauwhede (2009) concerning quasi-periodic stability of invariant tori with singular “normal” matrices in reversible systems.     

Partial preservation of frequencies and floquet exponents in KAM theory

Under a small perturbation of a completely integrable Hamiltonian system, invariant tori with Diophantine frequencies of motion are not destroyed but only slightly deformed, provided that the Hessian (with respect to the action variables) of the unperturbed Hamiltonian vanishes nowhere (the Kolmogorov nondegeneracy). The motion on every perturbed torus is quasiperiodic with the same frequencies. In this sense the frequencies of invariant tori of the unperturbed system are preserved. Recently, it has been found that the Kolmogorov nondegeneracy condition can be weakened so as to guarantee the preservation of only some subset of frequencies. Such partial preservation of frequencies can also be defined for lower dimensional invariant tori, whose dimension is less than the number of degrees of freedom. We consider a more general problem of partial preservation not only of the frequencies of invariant tori but also of their Floquet exponents (the eigenvalues of the coefficient matrix of the variational equation along the torus). The results are formulated for Hamiltonian, reversible, and dissipative systems (with a complete proof for the reversible case).     

Statistics of energy partitions for many-particle systems in arbitrary dimension

In some previous articles, we defined several partitions of the total kinetic energy T of a system of N classical particles in ? _d into components corresponding to various modes of motion. In the present paper, we propose formulas for the mean values of these components in the normalization T = 1 (for any d and N) under the assumption that the masses of all the particles are equal. These formulas are proven at the “physical level” of rigor and numerically confirmed for planar systems (d = 2) at 3 ? N ? 100. The case where the masses of the particles are chosen at random is also considered. The paper complements our article of 2008 [Russian J. Phys. Chem. B, 2(6):947–963] where similar numerical experiments were carried out for spatial systems (d = 3) at 3 ? N ? 100.     

The optimization of potential energy surface parameters for the CsCl + RbI system with the use of linear regression analysis

The influence of interaction potential parameters of likely charged ions on cross sections of various channels of a reaction of a pair of diatomic molecules with ionic bonds was studied in terms of quasi-classical trajectory simulation with the use of linear mean-square regressions. In the regression approach, the dependence of the cross section of a given reaction channel on potential parameters at each fixed collision energy is approximated by a linear function. We determined the region of softness parameters of the Cs⁺-Rb⁺ and Cl⁻-I⁻ interaction potentials. This region was optimum for the reproduction of experimental excitation functions of atomic and complex positive ions for the CsCl + RbI → products reaction.     

The structure of para-hydrogen clusters

The path integral Monte Carlo calculated radial distributions of para-hydrogen clusters $({\rm p}\text{-}{\rm H}_2)_N$ consisting of N = 4-40 molecules interacting via a Lennard-Jones potential at $T=1.5~{\rm K}$ show evidence for additional peaks compared to radial distributions calculated by diffusion Monte Carlo ( $T=0~{\rm K}$ ) and path integral Monte Carlo at $T \leq 0.5~{\rm K}$ . The difference in structures is attributed to quantum delocalization at the lowest temperature. The new structures at finite temperatures appear to be consistent with classical structures calculated for an effective Morse potential, which in order to account for the large zero point energy, is substantially softer than the Lennard-Jones potential.