Lie symmetries, perturbation to symmetries and adiabatic invariants of Lagrange system

For a generalized Hamiltonian system with the action of small forces of perturbation, the Lie symmetries, symmetrical perturbation, and adiabatic invariants is presented. Based on the invariance of equations of motion for the system under general infinitesimal transformation of Lie groups, the Lie symmetrical determining equations, and exact invariants of the system are given. Then the determining equations of Lie symmetrical perturbation and adiabatic invariants of the disturbed systems are obtained. Furthermore, in the special infinitesimal transformations, two deductions are given. At the end of the paper, one example is given to illustrate the application of the method and result.

A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems

In this paper, we present a new fractional dynamical theory, i.e., the dynamics of a Birkhoffian system with fractional derivatives (the fractional Birkhoffian mechanics), which gives a general method for constructing a fractional dynamical model of the actual problem. By using the definition of combined fractional derivative, we present a unified fractional Pfaff action and a unified fractional Pfaff–Birkhoff principle, and give four kinds of fractional Pfaff–Birkhoff principles under the different definitions of fractional derivative. And then, by using the fractional Pfaff–Birkhoff principles, we establish a series of fractional Birkhoffian equations with different fractional derivatives and construct the tensor representation of autonomous fractional Birkhoffian equations. Further, we study the relationship among the fractional Bikhoffian system, the fractional Hamiltonian system and the fractional Lagrangian system and give the transformation conditions. And furthermore, as applications of the fractional Birkhoffian method, we construct five kinds of fractional dynamical models, which include the fractional Lotka biochemical oscillator model, the fractional Whittaker model, the fractional Hojman–Urrutia model, the fractional Henon–Heiles model and the fractional Emden model.

Fractional Birkhoffian mechanics

Most of Quantum Secret Sharing(QSS) are (n, n) threshold 2-level schemes, in which the 2-level secret cannot be reconstructed until all n shares are collected. In this paper, we propose a (t, n) threshold d-level QSS scheme, in which the d-level secret can be reconstructed only if at least t shares are collected. Compared with (n, n) threshold 2-level QSS, the proposed QSS provides better universality, flexibility, and practicability. Moreover, in this scheme, any one of the participants does not know the other participants’ shares, even the trusted reconstructor Bob
 1 is no exception. The transformation of the particles includes some simple operations such as d-level CNOT, Quantum Fourier Transform(QFT), Inverse Quantum Fourier Transform(IQFT), and generalized Pauli operator. The transformed particles need not to be transmitted from one participant to another in the quantum channel. Security analysis shows that the proposed scheme can resist intercept-resend attack, entangle-measure attack, collusion attack, and forgery attack. Performance comparison shows that it has lower computation and communication costs than other similar schemes when 2 < t < n − 1.

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(t, n) Threshold d-Level Quantum Secret Sharing.

We evaluate the top-quark FCNC productions induced by the topcolor-assisted technicolor (TC2) model at the LHC. These productions proceed, respectively, through the parton-level processes gg -> t (c) over bar, cg -> t, cg -> tg, cg -> tZ, and cg -> t gamma. We show the dependence of the production rates on the relevant TC2 parameters and compare the results with the predictions in the minimal supersymmetric model. We find that for each channel the TC2 model allows for a much larger production rate than the supersymmetric model. All these rare productions in the TC2 model can be enhanced above the 3 sigma sensitivity of the LHC. Since in the minimal supersymmetric model only cg -> t is slightly larger than the corresponding LHC sensitivity, the observation of these processes will favor the TC2 model over the supersymmetric model. In case of unobservation, the LHC can set meaningful constraints on the TC2 parameters.

Top-quark FCNC Productions at CERN LHC in Topcolor-assisted Technicolor Model

For the rotational relativistic Hamiltonian system, a new type of the Lie symmetries and conserved quantities are given. On the basis of the theory of invariance of differential equations under infinitesimal transformations, and introducing infinitesimal transformations for generalized coordinates and generalized momentums , the determining equations of Lie symmetrical transformations of the system are constructed, which only depend on the canonical variables. A set of non-Noether conserved quantities are directly obtained from the Lie symmetries of the system. An example is given to illustrate the application of the results.

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

The definition and the criterion of Mei symmetry of generalized Hamilton systems with additional terms is studied in this paper. A necessary and sufficient condition for Mei symmetry of systems to be Lie symmetry is given. The Hojman conserved quantity for the differential equation of motion of generalized Hamilton system with additional terms being Mei symmetry can be deduced indirectly by Lie symmetry. An example is given to show the application of this result.

Mei symmetry of generalized Hamilton systems with additional terms

Hojman conserved quantities deduced by using the special Lie symmetry, the Noether symmetry and the Mei symmetry for systems with non-Chetaev nonholonomic constraints in the event space are studied. First, the differential equations of motion for the above systems are established. Second, the criterion of the Lie symmetry, the Noether symmetry, the Mei symmetry and the relation between them are obtained. Third, the conservation law obtained by Hojman is generalized and applied to the systems, and Hojman conserved quantities are obtained. An example is given to illustrate the application of the results.

Hojman conserved quantities for systems with non-Chetaev nonholonomic constraints in the event space

Special Lie?Mei symmetry and conserved quantities for Appell equations expressed by Appell functions in a holonomic mechanical system are investigated. On the basis of the Appell equation in a holonomic system, the definition and the criterion of special Lie?Mei symmetry of Appell equations expressed by Appell functions are given. The expressions of the determining equation of special Lie?Mei symmetry of Appell equations expressed by Appell functions, Hojman conserved quantity and Mei conserved quantity deduced from special Lie?Mei symmetry in a holonomic mechanical system are gained. An example is given to illustrate the application of the results.

Special Lie—Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function

This paper studies the Mei symmetry and Mei conserved quantity of non-Chetaev's type in the event space. The differential equations of motion of non-holonomic s ystems of non-Chetaev's type in event spaces are established. The definition and the criteria of Mei symmetry are given. The condition and the form of Mei conse rved quantity are deduced directly from the Mei symmetry. An example is given to illustrate the application of the results.

Jia Li-Qun

Papers

A Set of Lie Symmetrical Conservation Law for Rotational Relativistic Hamiltonian Systems

Mei symmetry of generalized Hamilton systems with additional terms

Hojman conserved quantities for systems with non-Chetaev nonholonomic constraints in the event space

Special Lie—Mei Symmetry and Conserved Quantities of Appell Equations Expressed by Appell Function

Mei symmetry and Mei conserved quantity of nonholonomic systems of non-Chetaev’s type in event space