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Volume 13, Number 5

Volume 13, Number 5, 2008
Nonholonomic mechanics + regular articles

Anikin A.
Normal Form of a Quantum Hamiltonian with One and a Half Degrees of Freedom Near a Hyperbolic Fixed Point
Abstract
According to classical result of Moser [1] a real-analytic Hamiltonian with one and a half degrees of freedom near a hyperbolic fixed point can be reduced to the normal form by a real-analytic symplectic change of variables. In this paper the result is extended to the case of the non-commutative algebra of quantum observables.We use an algebraic approach in quantum mechanics presented in [2] and develop it to the non-autonomous case. We introduce the notion of quantum non-autonomous canonical transformations and prove that they form a group and preserve the structure of the Heisenberg equation. We give the concept of a non-commutative normal form and prove that a time-periodic quantum observable with one degree of freedom near a hyperbolic fixed point can be reduced to a normal form by a canonical transformation. Unlike traditional results, where only formal theory of normal forms is constructed, we prove a convergence of the normalizing procedure.
Keywords: algebra of quantum observables, quantum normal forms, non-autonomous quantum dynamics
Citation: Anikin A., Normal Form of a Quantum Hamiltonian with One and a Half Degrees of Freedom Near a Hyperbolic Fixed Point, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 377-402
DOI:10.1134/S1560354708050018
Burdík C.,  Navrátil O.
New Formula for the Eigenvectors of the Gaudin Model in the $sl(3)$ Case
Abstract
We propose new formulas for eigenvectors of the Gaudin model in the sl(3) case. The central point of the construction is the explicit form of some operator $P$, which is used for derivation of eigenvalues given by the formula
$|w_1, w_2) = \sum\limits_{n=0}^\infty \frac{P^n}{n!}| w_1, w_2, 0>$,
where $w_1$, $w_2$ fulfil the standard well-know Bethe Ansatz equations.
Keywords: Gaudin model, Bethe Ansatz
Citation: Burdík C.,  Navrátil O., New Formula for the Eigenvectors of the Gaudin Model in the $sl(3)$ Case, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 403-416
DOI:10.1134/S156035470805002X
Nagoya H.,  Grammaticos B.,  Ramani A.
Quantum Painlevé Equations: from Continuous to Discrete and Back
Abstract
We present a cascade of quantum Painlevé equations consisting in successive contiguity relations, whereupon starting form a continuous equations we obtain a discrete one, and continuous limits of the latter. We start from the quantum Painlevé V and in the process derive the quantum form of continuous PIII which was missing in previous studies.
Keywords: discrete systems, quantization, Painlevé equations
Citation: Nagoya H.,  Grammaticos B.,  Ramani A., Quantum Painlevé Equations: from Continuous to Discrete and Back, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 417-423
DOI:10.1134/S1560354708050031
Novokshenov V. Y.
Zero-Dispersion Limit to the Korteweg-de Vries Equation: a Dressing Chain Approach
Abstract
An asymptotic solution of the KdV equation with small dispersion is studied for the case of smooth hump-like initial condition with monotonically decreasing slopes. Despite the well-known approaches by Lax–Levermore and Gurevich–Pitaevskii, a new way of constructing the asymptotics is proposed using the inverse scattering transform together with the dressing chain technique developed by A. Shabat [1]. It provides the Whitham-type approximaton of the leading term by solving the dressing chain through a finite-gap asymptotic ansatz. This yields the Whitham equations on the Riemann invariants together with hodograph transform which solves these equations explicitly. Thus we reproduce an uniform in $x$ asymptotics consisting of smooth solution of the Hopf equation outside the oscillating domain and a slowly modulated cnoidal wave within the domain. Finally, the dressing chain technique provides the proof of an asymptotic estimate for the leading term.
Keywords: KdV, small dispersion limit, wave collapse, dressing chain
Citation: Novokshenov V. Y., Zero-Dispersion Limit to the Korteweg-de Vries Equation: a Dressing Chain Approach , Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 424-430
DOI:10.1134/S1560354708050043
Kozlov V. V.
Gauss Principle and Realization of Constraints
Abstract
The paper generalizes the classical Gauss principle for non-constrained dynamical systems. For large anisotropic external forces of viscous friction our statement transforms into the common Gauss principle for systems with constraints.
Keywords: Gauss principle, constraints, anisotropic friction
Citation: Kozlov V. V., Gauss Principle and Realization of Constraints, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 431-434
DOI:10.1134/S1560354708050055
Ivanov A. P.
Geometric Representation of Detachment Conditions in Systems with Unilateral Constraint
Abstract
Mechanical systems with unilateral constraints that can be represented in the contact mode on the phase plane are considered. On the phase plane we construct domains that satisfy the following conditions 1) a detachment from the constraint is impossible; 2) the sign of the constraint reaction corresponds to its unilateral character. These conditions are equivalent for an ideal constraint [1, 2], but they can differ in the presence of friction [3]. Trajectories without detachments belong to intersections of these domains. A circular disc moving on a horizontal support with viscous friction and a disc with the sharp edge moving on an icy surface [4, 5] are considered as examples.
Usually for the control of contact conservation one uses only the second condition from above, which can lead to invalid qualitative conclusions.
Keywords: unilateral constraint, detachment conditions
Citation: Ivanov A. P., Geometric Representation of Detachment Conditions in Systems with Unilateral Constraint, Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 435-442
DOI:10.1134/S1560354708050067
Borisov A. V.,  Mamaev I. S.
Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems
Abstract
This paper can be regarded as a continuation of our previous work [1, 2] on the hierarchy of the dynamical behavior of nonholonomic systems. We consider different mechanical systems with nonholonomic constraints; in particular, we examine the existence of tensor invariants (laws of conservation) and their connection with the behavior of a system. Considerable attention is given to the possibility of conformally Hamiltonian representation of the equations of motion, which is mainly used for the integration of the considered systems.
Keywords: nonholonomic systems, implementation of constraints, conservation laws, hierarchy of dynamics, explicit integration
Citation: Borisov A. V.,  Mamaev I. S., Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems , Regular and Chaotic Dynamics, 2008, vol. 13, no. 5, pp. 443-490
DOI:10.1134/S1560354708050079

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