Lie symmetries and conserved quantities of controllable nonholonomic dynamical systems

This paper concentrates on studying the Lie symmetries and conserved quantities of controllable nonholonomicdynamicM systems. Based on the infinitesimal transformation, we establish the Lie symmetric determining equationsand restrictive equations and give three definitions of Lie symmetries before the structure equations and conservedquantities of tile Lie symmetries are obtained. Then we make a study of the inverse problems. Finally, an example ispresented for illustrating the results.     

Noncommutative *-product continual Lie algebras and solvable models

Noncommutative counterparts of exactly solvable models are proposed on the basis of *-product continual Lie algebras. Examples of noncommutative Liouville and sine/sh-Gordon equations are discussed.     

Time-Dependent First Integrals,Nonlinear Dynamical Systems,and Numerical Integration

Many nonlinear dynamical systems expressed as autonomous systems of first-order ordinary differential equations admit first integrals and explicitly time-dependent first integrals. Under numerical integration these first integrals should be preserved. We discuss this case for explicitly time-dependent first integrals.     

Integrable systems on so(n)-valued Lie algebras on higher genus curves

We consider special quasi-graded so(n)-valued Lie algebras on higher genus curves. Using them we find new Lie theoretical interpretation of the integrability of generalized Manakov tops and generalized Clebsch systems and explicitly construct their isomorphism on trajectories.     

Lie symmetry algebra of one-dimensional nonconservative dynamical systems

Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1--1 mapping, the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion.     

SUSY QM, symmetries and spectrum generating algebras for two-dimensional systems

We show in a systematic and clear way how factorization methods can be used to construct the generators for hidden and dynamical symmetries. This is shown by studying the 2D problems of hydrogen atom, the isotropic harmonic oscillator and the radial potential Aρ^2ζ−2 − Bρ^ζ−2. We show that in these cases the non-compact (compact) algebra corresponds to so(2, 1) (su(2)).     

On the study of unitary representations of the twisted Heisenberg-Virasoro algebra via highest weight modules over affine Lie algebras

In this paper, we first construct an analogue of the Sugawara operators for the twisted Heisenberg-Virasoro algebra. By using these operators, we show that every integrable highest weight module over an affine Lie algebra can be viewed as a unitary representation of the twisted Heisenberg-Virasoro algebra. As a by-product of our constructions, we give the unitary representations of the twisted Heisenberg-Virasoro algebra which have the central charges appearing in [1]. Our approach to obtain these central charges is different with that of [1].     

Discrete variational principle and first integrals for Lagrange--Maxwell mechanico-electrical systems

This paper presents a discrete variational principle and a method to build first-integrals for finite dimensional Lagrange--Maxwell mechanico-electrical systems with nonconservative forces and a dissipation function. The discrete variational principle and the corresponding Euler--Lagrange equations are derived from a discrete action associated to these systems. The first-integrals are obtained by introducing the infinitesimal transformation with respect to the generalized coordinates and electric quantities of the systems. This work also extends discrete Noether symmetries to mechanico-electrical dynamical systems. A practical example is presented to illustrate the results.     

Invariant functions and contractions of certain types of Lie algebras of lower dimensions

In this paper, we deal with contractions of Lie algebras. We use two invariant functions of Lie algebras as a tool, named ψ and ? function, respectively, which have a great application in Physics due to their remarkable properties. We focus the study of these functions in the frame of the filiform Lie algebras, trying to extend to these algebras some of the properties of such functions over semi-simple Lie algebras.     

Dissipative finite degrees of freedom dynamical system and description of optical systems with saturable amplification, saturable losses and filtering

Single-mode fiber optical system with saturable amplification, saturable losses and spectral filtering as proposed by Rozanov and Fedorov (1998) [10] is studied. The system of ordinary differential equations (ODE’s) that can help investigation of the original physical system is proposed. It allows calculation of linear and nonlinear fixed points as well as the study of their stability, so it can be used for analysis of coherent structures and their classification. Derived system of ODE’s extends the earlier one proposed by van Saarloos and Hohenberg (1992) [2], for the analysis of coherent structures of the qubic-quintic Ginzburg-Landau equation, by including additionally the temporal dependences of the gain and losses. In order to verify it, it was applied to the earlier considered cases of fast and slow changes in the amplification and losses. Earlier obtained localized structures namely pulses, have been observed via numerical solution of the proposed system. In addition, new families of fronts have been identified.     

黑克、布劳、及伯曼-温采尔代数的诱导及分导系数和量子经典李代数的耦合与重新耦合系数

本文总结了计算黑克、布劳、及伯曼 温采尔代数在各种工数链下诱导及分导系数的线性方程方法(LEM)。特别强调了关于A,B,C,D型李代数及其量子情形与其中心代数之间的舒尔 魏尔 布劳双关性关系。这一关系使我们能够利用相应中心代数的诱导及分导系数计算出经典李代数及其量子情形的耦合与重新耦合系数。讨论了从该方法得到B,C,D型李代数不可约表示克罗内克积分解的应用。基于LEM还得到了处理对应于置换群CG系列问题的黑克代数张量积的方法。     

Flux Hamiltonians, Lie algebras, and root lattices with minuscule decorations

We study a family of Hamiltonians of fermions hopping on a set of lattices in the presence of a background gauge field. The lattices are constructed by decorating the root lattices of various Lie algebras with their minuscule representations. The Hamiltonians are, in momentum space, themselves elements of the Lie algebras in these same representations. We describe various interesting aspects of the spectra, which exhibit a family resemblance to the Dirac spectrum, and in many cases are able to relate them to known facts about the relevant Lie algebras. Interestingly, various realizable lattices such as the kagomé and pyrochlore can be given this Lie algebraic interpretation, and the particular flux Hamiltonians arise as mean-field Hamiltonians for spin-1/2 Heisenberg models on these lattices.     

Phase-coupled nonlinear dynamical systems, stability, first integrals, and Liapunov exponents

The stability of a class of coupled identical autonomous systems of first-order nonlinear ordinary differential equations is investigated. These couplings play a central role in controlling chaotic systems and can be applied in electronic circuits and laser systems. As applications we consider a coupled van der Pol equation and a coupled logistic map. When the uncoupled system admits a first integral we study whether a first integral exists for the coupled system. Gradient systems and the Painlevé property are also discussed. Finally, the relation of the Liapunov exponents of the uncoupled and coupled systems are discussed.     

Affine Lie algebras, string junctions and 7-branes

We consider the realization of affine ADE Lie algebras as string junctions on mutually non-local 7-branes in Type 1113 string theory. The existence of the affine algebra is signaled by the presence of the imaginary root junction δ, which is realized as a string encircling the 7-brane configuration. The level k of an affine representation partially constrains the asymptotic (p, q) charges of string junctions departing the configuration. The junction intersection form reproduces the full affine inner product, plus terms in the asymptotic charges.     

Nonlinear dynamical symmetries of Smorodinsky–Winternitz and and Fokas–Lagerstorm systems

General solutions of the Smorodinsky–Winternitz system and the Fokas–Lagerstorm system, which are superintegrable in two-dimensional Euclidean space, are obtained using the algebraic method (structure function). Their dynamical symmetries, which are governed by deformed angular momentum algebras, are revealed.