Symmetries of some classes of dynamical systems

In this paper a three-dimensional system with five parameters is considered. For some particular values of these parameters, one finds known dynamical systems. The purpose of this work is to study some symmetries of the considered system, such as Lie-point symmetries, conformal symmetries, master symmetries and variational symmetries. In order to present these symmetries we give constants of motion. Using Lie group theory, Hamiltonian and bi-Hamiltonian structures are given. Also, symplectic realizations of Hamiltonian structures are presented. We have generalized some known results and we have established other new results. Our unitary presentation allows the study of these classes of dynamical systems from other points of view, e.g. stability problems, existence of periodic orbits, homoclinic and heteroclinic orbits.     

Generalised Symmetries and the Ermakov-Lewis Invariant

Abstract

Generalised symmetries and point symmetries coincide for systems of first-order ordinary differential equations and are infinite in number. Systems of linear first-order ordinary differential equations possess a generalised rescaling symmetry. For the system of first-order ordinary differential equations corresponding to the time-dependent linear oscillator the invariant of this symmetry has the form of the famous Ermakov-Lewis invariant, but in fact reveals a richer structure.     

Killing and Noether Symmetries of Plane Symmetric Spacetime

This paper is devoted to investigate the Killing and Noether symmetries of static plane symmetric spacetime. For this purpose, five different cases have been discussed. The Killing and Noether symmetries of Minkowski spacetime in cartesian coordinates are calculated as a special case and it is found that Lie algebra of the Lagrangian is 10 and 17 dimensional respectively. The symmetries of Taub’s universe, anti-deSitter universe, self similar solutions of infinite kind for parallel perfect fluid case and self similar solutions of infinite kind for parallel dust case are also explored. In all the cases, the Noether generators are calculated in the presence of gauge term. All these examples justify the conjecture that Killing symmetries form a subalgebra of Noether symmetries (Bokhari et al. in Int. J. Theor. Phys. 45:1063, 2006).     

Symmetries of Maxwell-Bloch Equations

Abstract

We study symmetries of the real Maxwell-Bloch equations. We give a Lax pair, bi-Hamiltonian formulations and we find a symplectic realization of the system. We have also constructed a hierarchy of master symmetries which is used to generate nonlinear Poisson brackets. In addition we have calculated the classical Lie point symmetries and variational symmetries.     

Symmetries of sub-Riemannian surfaces

We obtain some results on symmetries of sub-Riemannian surfaces. In case of a contact sub-Riemannian surface we base on invariants found by Hughen [15]. Using these invariants, we find conditions under which a sub-Riemannian surface does not admit symmetries. If a surface admits symmetries, we show how invariants help to find them. It is worth noting, that the obtained conditions can be explicitly checked for a given contact sub-Riemannian surface. Also, we consider sub-Riemannian surfaces which are not contact and find their invariants along the surface where the distribution fails to be contact.     

How to Find Discrete Contact Symmetries

Abstract

This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism of the Lie algebra of generators of Lie contact symmetries. Consequently, all contact symmetries satisfy various compatibility conditions. These conditions enable the discrete symmetries to be found systematically, with little effort.     

Lie Symmetries, Perturbation to Symmetries and Adiabatic Invariants of a Generalized Birkhoff System

We study the perturbation to symmetries and adiabatic invariants of a generalized Birkhoff system. Based on the invariance of differential equations under infinitesimal transformations, Lie symmetries, laws of conservations, perturbation to the symmetries and adiabatic invariants of the generalized Birkhoff system are presented. First, the concepts of Lie symmetries and higher order adiabatic invariants of the generalized Birkhoff system are proposed. Then, the conditions for the existence of the exact invariants and adiabatic invariants are proved, and their forms are given. Finally, an example is presented to illustrate the method and results.     

Mei Symmetries and Lie Symmetries for Nonholonomic Controllable Mechanical Systems with Relativistic Rotational Variable Mass

The Mei symmetries and the Lie symmetries for nonholonomic controllable mechanical systems with relativistic rotational variable mass are studied. The differential equations of motion of the systems are established. The definition and criterion of the Mei symmetries and the Lie symmetries of the system are studied respectively. The necessary and sufficient condition under which the Mei symmetry is Lie symmetry is given. The condition under which the Mei symmetries can be led to a new kind of conserved quantity and the form of the conserved quantity are obtained. An example is given to illustrate the application of the results.     

Symmetries of Taub-NUT Dual Metrics

In this paper we study the symmetries of thedual Taub-nut metrics. Generic and non-genericsymmetries of dual Taub-nut metrics are investigated.The existence of the Runge-Lenz type symmetry isanalyzed for dual Taub-nut metrics. We find that in somecases the symmetries of the dual metrics are the samewith the symmetries of Taub-nut metric.     

∧超核中单粒子谱的自旋及赝自旋对称性研究(英文)

原子核中单粒子谱的自旋和赝自旋对称性对核结构的研究具有重要意义。本文基于相对论平均场理论,以132Sn,_∧133Sn,及_2∧134Sn为例,研究了∧超核中单∧谱和单中子谱的自旋和赝自旋对称性。研究发现,单∧谱的自旋对称性保持得相当好,与实验观测一致;而其赝自旋对称性只是近似保持,与核子谱的情况类似。此外,还研究了∧超子对中子谱的杂质效应,发现∧超子使中子谱的自旋对称性变差,赝自旋对称性变好。Spin and pseudospin symmetries in the single-particle spectra of atomic nuclei are of great significance for the study of nuclear structure. In this work, taking 132Sn, _∧133Sn, and _2∧134Sn as examples, the spin and pseudospin symmetries in ∧ hypernuclei are studied by using the relativistic mean-field model. For the single-∧ spectra, results show that the spin symmetry maintains well while the pseudospin symmetry is approximately conserved. Besides, as impurities, the ∧ hyperons worsen the spin symmetry of single-neutron spectra while improve the pseudospin symmetry.     

Classical Canonical General Coordinate and Gauge Symmetries

Classical generators of one-dimensional reparametrization, and higher dimensional diffeomorphism symmetries are displayed for the relativistic free particle, relativistic particles in interaction, and general relativity in both Lagrangian and Hamiltonian frameworks. Projectability of these symmetries under the Legendre map is achieved only with dynamical variable-dependent transformations. When gauge symmetries are included, as in Einstein-Yang-Mills and a new reparametrization covariant pre-Maxwell model, pure coordinate symmetries are not projectable. They must be accompanied by internal gauge transformations.     

Lie Algebras of Approximate Symmetries

Abstract

Properties of approximate symmetries of equations with a small parameter are discussed. It turns out that approximate symmetries form an approximate Lie algebra. A concept of approximate invariants is introduced and the algorithm of their calculating is proposed.     

Lie Point Symmetries of Differential-Difference Equations

In this paper, the classical Lie group approach is extended to find some Lie point symmetries of differentialdifference equations. It reveals that the obtained Lie point symmetries can constitute a Kac-Moody-Virasoro algebra.     

Symmetries in inelastic proton-nucleus scattering

Various symmetries of inelastic proton-nucleus scattering within the Glauber model are discussed. They are obtained from symmetries of the corresponding proton-nucleon amplitude. In particular, it is shown that the polarization and analyzing power are in general not equal.     

Hopf-Algebra Description of Noncommutative-Spacetime Symmetries

A brief summary is given of the results reported in [hep-th/0306013], in collaboration with G. Amelino-Camelia and F. D'Andrea. It is focused on the analysis of the symmetries of -Minkowski noncommutative spacetime, described in terms of a Weyl map. The commutative-spacetime notion of Lie-algebra symmetries must be replaced by the one of Hopf-algebra symmetries. However, in the Hopf-algebra sense, it is possible to construct an action in -Minkowski, which is invariant under a 10-generators Poincaré-like symmetry algebra.     

Symmetries on quantum logics

We study conditions under which the group of symmetries of a quantum logic is isomorphic to the group of symmetries on certain subsets of the state space of the logic. The notions of Jordan–Hahn decomposition and ultrafulness of the set of states under consideration play a fundamental role in these investigations. They are used to establish a connection between the elements of the logic and the weak¹-exposed points or extreme points of the unit interval of the Banach dual of the signed state space. The results are then interpreted in the standard logic of quantum mechanics.     

Noether Symmetries Versus Killing Vectors and Isometries of Spacetimes

Symmetries of spacetime manifolds which are given by Killing vectors are compared with the symmetries of the Lagrangians of the respective spacetimes. We find the point generators of the one parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian (Noether symmetries). In the examples considered, it is shown that the Noether symmetries obtained by considering the Larangians provide additional symmetries which are not provided by the Killing vectors. It is conjectured that these symmetries would always provide a larger Lie algebra of which the KV symmetres will form a subalgebra. PACS: 04.25.-g, 02.20.Sv, 11.30.-j     

Quaternion-Octonion Unitary Symmetries and Analogous Casimir Operators

An attempt has been made to investigate the global SU(2) and SU(3) unitary flavor symmetries systematically in terms of quaternion and octonion respectively. It is shown that these symmetries are suitably handled with quaternions and octonions in order to obtain their generators, commutation rules and symmetry properties. Accordingly, Casimir operators for SU(2) and SU(3) flavor symmetries are also constructed for the proper testing of these symmetries in terms of quaternions and octonions.